Journal of Mathematical Sciences, Vol. 162, No. 6, 2009
DYNAMICAL SYSTEMS WITH VARIABLE DISSIPATION: APPROACHES, METHODS, AND APPLICATIONS M. V. Shamolin
UDC 517.925+531.01+531.552
Abstract. This work is devoted to the development of qualitative methods in the theory of nonconservative systems that arise, e.g., in such fields of science as the dynamics of a rigid body interacting with a resisting medium, oscillation theory, etc. This material can arouse the interest of specialists in the qualitative theory of ordinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics since the work uses the properties of motion of a rigid body in a medium under streamline flow-around conditions. The author obtains a full spectrum of complete integrability cases for nonconservative dynamical systems having nontrivial symmetries. Moreover, in almost all cases of integrability, each of the first integrals is expressed through a finite combination of elementary functions and is a transcendental function of its variables, simultaneously. In this case, the transcendence is meant in the complex analysis sense; i.e., after the continuation of the functions considered to the complex domain, they have essentially singular points. The latter fact is stipulated by the existence of attracting and repelling limit sets in the system considered (for example, attracting and repelling foci). The author obtains new families of phase portraits of systems with variable dissipation on lowerand higher-dimensional manifolds. He discusses the problems of their absolute or relative roughness, He discovers new integrable cases of rigid body motion, including those in the classical problem of motion of a spherical pendulum placed in an over-running medium flow.
CONTENTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. A Certain Problem of the Dynamics of a Rigid Body Interacting with a Medium . . . . . . 1.1. From Historical Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Some Modern Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Sequence of Steps in Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Physical Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Linearized Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Beginning of Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Directions Developed in the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. Brief Contents of Other Sections of Work . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Complete Integrability of Certain Classes of Nonconservative Systems . . . . . . . . . . . . 2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Variable Dissipation Dynamical Systems and Their General Properties . . . . . . . . . 2.3. One of the Definitions of Systems with Zero Mean Variable Dissipation . . . . . . . . . 2.4. Systems with Symmetries and Zero Mean Variable Dissipation . . . . . . . . . . . . . . 2.5. Systems on S1 {α mod 2π} × R1 {ω} or Two-Dimensional Cylinder . . . . . . . . . . . . 2.6. Systems on S1 {α mod 2π} \ {α = 0, α = π} × R2 {z1 , z2 } or Tangent Bundle of Two-Dimensional Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Systems from Dynamics of a Four-Dimensional Rigid Body Interacting with a Medium
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 14, No. 3, pp. 3–237, 2008. c 2009 Springer Science+Business Media, Inc. 1072–3374/09/1626–0741
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3. Class of Generating Dynamical Systems on Lower-Dimensional Manifolds: Once More about Resisting Medium Action on a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Methodology for Finding Unknown Dimension-Free Parameters of Medium Action on a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Nonlinear Dynamical Systems Describing Various Variants of Body Motion in a Medium 4. Some Problems of Qualitative Theory of Ordinary Differential Equations in Variable Dissipation System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Remarks on Bifurcation of Birth of Poincar´e–Andronov–Hopf Cycle . . . . . . . . . . . . 4.2. On Closed Curves of Trajectories Contractible to a Point along Phase Surface . . . . . . 4.3. On the Absence of Closed Curves of Trajectories Not Contractible to a Point along the Phase Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. On Existence of Poincar´e Topographic Systems in Dynamics of a Rigid Body Interacting with a Resisting Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Contact Curves and Comparison Systems. Remarks on Limit Cycles and Problem of Distinguishing Center and Focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. On Trajectories Having Infinitely Distant Points of the Plane as Limit Sets . . . . . . . . 4.7. Periodic and Poisson Stable Trajectories in Phase Spaces of Dynamical Systems . . . . . 4.8. Spatial Poincar´e Topographical Systems and Comparison Systems . . . . . . . . . . . . . 5. Relative Structural Stability and Relative Structural Instability of Various Degrees . . . . . . 5.1. Definition of Relative Structural Stability (Relative Roughness) . . . . . . . . . . . . . . 5.2. Relative Structural Instability (Relative Nonroughness) of Various Degrees . . . . . . . . 5.3. Examples from Dynamics of a Rigid Body Interacting with a Medium and Oscillation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Methods for Analyzing Zero Mean Variable Dissipation Dynamical Systems in Rigid Body Plane Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Case of Body Motion in a Medium under a Certain Nonintegrable Constraint and Beginning of Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. On Transcendental Integrability of System . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. On Mechanical Analogy with a Pendulum in Medium Flow . . . . . . . . . . . . . . . . . 6.4. Topological Structure of Phase Portrait of System Studied . . . . . . . . . . . . . . . . . 6.5. General Properties of Solutions of Dynamical System . . . . . . . . . . . . . . . . . . . . 6.6. Fiberings of Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Properties of Solutions Corresponding to Oscillatory Domain . . . . . . . . . . . . . . . . 6.8. Properties of Solutions Corresponding to Rotational Domain . . . . . . . . . . . . . . . . 6.9. On Technical Tools for Model Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10. Reduction of System to a Physical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . 6.11. Beginning of Qualitative Analysis. Equilibrium Points of Systems and Stationary Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12. Bundles of Phase Space, Its Symmetry, and Beginning of Topological Analysis . . . . . . 6.13. On Existence of an Additional Transcendental Integral . . . . . . . . . . . . . . . . . . . 6.14. Topological Structure of Phase Portraits of System on a Two-Dimensional Cylinder . . . 6.15. Mechanical Interpretation of Some Singular Phase Trajectories . . . . . . . . . . . . . . . 7. Methods for Analyzing Nonzero Mean Variable Dissipation Dynamical Systems in Plane Dynamics of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Beginning of Qualitative Analysis. Equilibrium Points of Second- and Third-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Bundles of Phase Space, Its Symmetries, and Beginning of Topological Analysis . . . . . 7.3. Classification of System Phase Portraits on a Two-Dimensional Cylinder for the First Parameter Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
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7.4. A Classification of Portraits for Second and Third Parameter Domains . . . . . . . . . . 7.5. Structure of System Phase Portrait in the Fourth Parameter Domain . . . . . . . . . . . 8. Methods for Analyzing Zero Mean Variable Dissipation Dynamical Systems in Spatial Dynamics of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Statement of Problem of Spatial Body Motion in a Resisting Medium under Streamline Flow Around . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Case of Body Motion in a Medium under a Certain Nonintegrable Constraint and Beginning of Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. On Transcendental Integrability of the System . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Problem on Spatial Pendulum in an Over-Running Medium Flow . . . . . . . . . . . . . 8.5. Topological Structure of Phase Portrait of System Studied . . . . . . . . . . . . . . . . . 8.6. Trajectories of Spherical Pendulum Motion and the Case of Its Nonzero Twist near the Longitudinal Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Methods for Analyzing Nonzero Mean Variable Dissipation Dynamical Systems in Rigid Body Spatial Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Case of Zero Longitudinal Component of Angular Velocity and Corresponding Stationary Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Bundles of Phase Space, Its Symmetries, and Beginning of Topological Analysis . . . . . 9.3. Classification of System Phase Portraits in Three-Dimensional Space for a Certain Parameter Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Some Problems of Plane Dynamics of a Rigid Body Interacting with a Medium in the Presence of Linear Medium Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Body Free Drag in a Medium Taking Account of Linear Damping Moment . . . . . . . . 10.2. Motion in a Medium in the Presence of a Certain Nonintegrable Constraint and Linear Damping Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Topological Structure of Some Phase Portraits in Problem of Body Motion in a Medium Taking Account of Damping Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4. Comparisons of Some Classes of Body Motions in a Medium in the Absence and Presence of Linear Damping Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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To my mother Shamolina Tamara Nicolaevna Science is always wrong, it never solves a problem without creating ten more. George Bernard Shaw
Introduction This work is devoted to the development of qualitative methods in the theory of nonconservative systems that arise, e.g., in such fields of science as the dynamics of a rigid body interacting with a resisting medium, oscillation theory, etc. This material can arouse the interest of specialists in the qualitative theory of ordinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics since the work uses the properties of motion of a rigid body in a medium under streamline flow-around conditions. The author obtains a full spectrum of complete integrability cases for nonconservative dynamical systems having nontrivial symmetries. Moreover, in almost all cases of integrability, each of the first integrals is expressed through a finite combination of elementary functions and is a transcendental function 743
of its variables, simultaneously. In this case, the transcendence is meant in the complex analysis sense; i.e., after the continuation of the functions considered to the complex domain, they have essentially singular points. The latter fact is stipulated by the existence of attracting and repelling limit sets in the system considered (for example, attracting and repelling foci). The author obtains new families of phase portraits of systems with variable dissipation on lowerand higher-dimensional manifolds. He discusses the problems of their absolute or relative roughness. He discovers new integrable cases of rigid body motion, including those in the classical problem of motion of a spherical pendulum placed in an over-running medium flow. 1. A Certain Problem of the Dynamics of a Rigid Body Interacting with a Medium 1.1. From Historical Past. The problem of motion of a rigid body in a resisting medium (for example, the problem of body fall in air) has aroused the interest of investigators for several hundred years: as far back as in the Middle Ages, there arose the necessity to study the dependence of the firing range on the elevation angle of the gun barrel. The trials of studying the motion of a body in the air and in a fluid allowed Ch. Huygens to discover the empirical law saying that the resistance is proportional to the square of the body motion velocity in the air (1669). Basing on the trials of Ph. Hoxby, J. Desaguliers, and his own, Isaac Newton created the mathematical theory of resistance of the air; in the 18th century, its elaboration was continued by Varingnon, D. Bernoulli, J. D’Alembert, L. Euler, and others. The ballistic pendulum was invented at that time. As a result of a deep analysis of the trial material belonging to Englishman B. Robbins, in 1745, Leonard Euler replaced the quadratic resistance law by the following binomial law: the first summand is proportional to the velocity square, and the second to the velocity fourth power. Subsequently, Euler elaborated numerical methods for integrating the differential equation of the shell motion, in particular, using slow convergent series. For the target shooting, he suggested another methodology according to which the shell motion is partitioned into components one of which is related to the resistance. The efforts of the scientists were aimed not only at finding the shell trajectory and the shell motion law but also at the maximally possible account of additional phenomena that leads to the important corrections to the basic theory. In the 18th century, Robbins observed that the center of mass of a rotating shell describes a spatial curve. Later on, in the 19th century, S. Poisson and then M. V. Ostrogradskii tried to give a mathematical treatment of this phenomenon. Basing on the general theory of the rigid body motion, it was found that an oblong rotating shell obeys a proper rapid rotation around the longitudinal axis of dynamical and geometric symmetry, the precession near the shell velocity vector, and a nutational motion near the tilting momentum vector. 1.1.1. Studies of N. E. Zhukovskii and S. A. Chaplygin. It was Zhukovskii who analyzed various problems of the point dynamics in a medium, more precisely, the body fall, the motion of a body moving at an angle with respect to the horizon, the pendulum motion, etc. Along with integrating the equations of motion, he improved the model of the body interaction with a resisting medium and considered that the kinetic energy of a falling body is spent in the formation of vortex air motions and, in addition, on overcoming the molecular forces of the air adhering to the moving body. The resistance depends not only on the velocity of the body point motion but on the form of the body itself. If the velocity is small, then with sufficient accuracy, one can assert that the resistance is proportional to the first power of the velocity. For large velocities, the resistance is proportional to the velocity square. Also, from the Zhukovskii studies, an attempt is known to model the motion based on experiments in self-rotations of plates falling in the air [457, 458] (the so-called “Hamburg cardboard”). Here one needs to take account of such properties of action of the medium on the body as the resistance force and the body force. Precisely, the aerodynamic characteristics of a plate are also used for modelling bird flight [458]. 744
Zhukovskii assumed the existence of a “bird body” in dynamic equilibrium with respect to the center of mass such that the angle between the center-of-mass velocity and the plate wing plane (angle of attack) serves as a control parameter, i.e., it can be given arbitrarily. This assumption is equivalent to the assumption on such a body motion separation under which the characteristic time of motion with respect to the center of mass is considerably smaller than the characteristic times of motion of the center itself. The study of the body motion in a medium under the condition that its translational motion is connected with the rotational motion is of interest. The problems mentioned above are far from exhaustive for all possibilities of such a type. Among the studies of Chaplygin, we also mention the statement of the problem of heavy body motion in an incompressible fluid [85, 86]. In the framework of this work, our fundamental problem is the study of the infinite length plate motion under streamline flow-around conditions [85]. First of all, this problem is important for further studying the motion of a body interacting with a medium through the frontal plane part. 1.1.2. Various aspects of the problem. As is seen, in the historical past, only one aspect of the problem of the body motion in a resisting medium was considered. Precisely, the interests of investigators were aimed at obtaining concrete trajectories, although in an approximate but explicit form. Moreover, in parallel, the problem of more precise modelling of the interaction of a body with a resisting medium was considered. See also [7, 54, 177, 190, 199, 211, 214, 215, 231, 454] for interesting experimental phenomena. Let us briefly illustrate the above problem by examining bodies of a simple form. A plane plate is the simplest body that allows one to study various features of motion in a medium. The effects related to the influence of adhered masses (classical Kirchhoff problem) are demonstrated in [168] by examining the motion of a body-plate in a fluid (as is known, the study was initiated by Thomson, Tate, and Kirchhoff). The Kirchhoff problem posed in the second half of the 19th century opened the second aspect of the problem. It is related to the integrability problems for a system of differential equations [168] that describes this motion (the problems of existence of analytic (smooth, meromorphic) first integrals). Up to the present, because of complexity, various variants of the Kirchhoff problem were almost always considered from the viewpoint of the integrability problem, and only in certain cases was the qualitative analysis of a number of trajectories carried out. In the works of Kirchhoff, Clebsch, Steklov, Chaplygin, Lyapunov, Kharlamov, etc., some conditions for existence of an additional analytic first integral were found. At present, the solution of the problem has been improved: in [219] the theory of integrable cases (constructing L-A-pair) is created, whereas in [163] conditions for nonexistence of an additional first integral of the Kirchhoff equations are found (see also [26, 57–59, 76, 77, 206, 234, 235, 435, 450]). Also, let us mention the third aspect of the above problem, precisely, the qualitative analysis of the systems of differential equations describing this motion (phase space fiberings, the qualitative location of phase trajectories, symmetries, etc.). Although the listed problems are closely related to the integrability, their solution is of an independent nature. Moreover, this aspect stimulates the development of qualitative means. 1.2. Some Modern Results. The problem of modelling rigid body motion in a medium is based on a possible model restriction of the problem considered and developing mathematical tools. So, in [158,159], the authors study the Chaplygin problem on the fall of a heavy body having a symmetry plane in an infinite volume of an ideal fluid. In this case, they analyze the properties of trajectories, whereas in [154, 155], the author characterizes the properties of Bernoulli surfaces of constant curvature for a compressible continuous medium. Under general assumptions on the character of the aerodynamic action, in [178–181], the author studies the existence and stability problems of stationary motion regimes in a medium. The problem of stability of the body’s permanent rotation in the medium flow (auto-rotation regime; see [230] and 745
also [63]) is also of interest. A special structure of the body surface and the hypothesis about the quasistatic medium action allows the authors to formulate the complete force scheme containing the mass, geometric, and aerodynamic characteristics. The auto-rotation regime and its stability are studied. The Magnus effect is modelled whose nonconservative character considerably influences the stability of the body rotation in a medium. The Raleigh function theory is well demonstrated in studying the fall of a heavy homogeneous plate of rectangular form in a resisting medium [158, 159]. Together with the viscous resistance given by the Raleigh function, the adhered mass effect is taken into account. In the framework of the atmosphere flight theory of an axially-symmetric rotating rigid body, the problem of maximizing the flight distance is studied [178–181]. The angular velocity of the body rotation around the symmetry axis is considered as a control. Interesting interaction models are highlighted in [41–44]. The influence of the aerodynamic forces on the satellite rotation and orientation is clearly justified from the viewpoint of the qualitative theory. The main effects of the satellite rotational motion dynamics under the action of moments, including aerodynamic ones, are considered in [43], whereas the dynamics of the celestial body rotational motion in the gravitational fields is considered in [42] with special account of the resonance effects. Let us dwell on the work [209] in more detail. In this work, the authors construct model dynamical systems that allow one to study the motion of the center of mass for a dynamically-symmetric body of a spatial aerodynamical form with higher supporting properties under a nonstationary flight. In the framework of the quasi-stationary linearized model of the aerodynamic action that does not take into account the influence of the damping moments of the aerodynamic forces, the authors reveal the damping influence of the body force and find the restrictions on the aerodynamic coefficients under which the effective damping of angular body oscillations is ensured. Under high-velocity flight conditions when the aerodynamic action on the body considerably exceeds the influence of the gravity force, they obtain an analytical solution of the nonstationary dynamical system linearized in a part of the variables, which describes the body motion with respect to the center of mass. The tools for obtaining the described results are strictly discussed in [210], where the author elaborates a library of applied programs that ensure the multi-window representation of the graphical information about the behavior of various components of the state vector of the dynamical model considered. This series of works was started about ten years ago [208], and at present, it is being developed at the Laboratory of Navigation and Control at the Institute of Mechanics of M. V. Lomonosov Moscow State University. 1.3. Sequence of Steps in Modelling. Generally speaking, the general problem of studying the body motion in the resistance force field “is prevented” by the absence of any complete description of this force field. As is known, in principle, we can measure the positional component of the resistance force in a stationary experiment. But the component of the force field, which corresponds to the quasi-velocities of the system considered, arises only under the nonstationary body motion. Therefore, the process of describing the force field is a sequence of steps. First we study a preparatory model of the force field and construct a family of mechanical systems whose motion has different characteristics that essentially depend on model parameters such that the information about them is incomplete or does not exist at all. As a result of studying such a model, there arise questions such that the answers to them cannot be found in the framework of the accepted model. Then the elaborated objects become the subject of a detailed experimental study at the second step. Such an experiment either presupposes the answers to the formulated questions and introduces necessary corrections to the preparatory constructed model or reveals new questions, which lead to the necessity of repeating the first step but at a new level of understanding of the problem. Such an approach is related to the description of stationary motion regimes, their branching, bifurcation, stability and instability analysis, revealing surgery conditions, and appearance of regular or irregular (i.e., chaotic) oscillations. 746
Sometimes, we may succeed in obtaining the answers to questions of qualitative character when discussing the traditional problem of analytic mechanics, the existence problem of the full tuple of first integrals for the constructed dynamical system. At the same time, the study of the behavior of a dynamical system “as a whole” often forces us to use a numerical experiment. In this case, there arises the necessity of elaborating new computational algorithms or improving the known, as well as new qualitative methods. In this work, we study the problem on the body motion under the condition that the line of the force applied to the body does not change its orientation with respect to the body and can only displace parallel to itself depending on the angle of attack and, possibly, on other phase variables. Such conditions arise under the plate motion with the so-called “large” angles of attack in a medium under a streamline flow around (in this case, the fluid is assumed to be ideal in general, although all this is also true for fluids of small viscosity, first of all, for water) [85, 86, 141, 253] or under a separation flow around [441] (which is justified by an experiment as completely satisfactory). Therefore, the main object of studying is a family of bodies such that a part of the surface of each has a plane part where a medium flows around according to the streamline flow-around laws. The mathematical model of the rigid body motion used here was previously analyzed. So, in [107,182, 184, 185, 244], the authors construct the phase portrait of the physical pendulum placed in the medium flow. The dynamical system describing the pendulum motion exhibits nontrivial nonlinear properties, which makes necessary a further complete nonlinear analysis and a possible creation of the methodology for studying. In [105, 242], the authors examine the stability problem of free body rectilinear motions under a streamline flow-around. The study is carried out on the basis of the linearized equations for the body motion. Therefore, following [107–109, 243, 245], first we describe the linear model more concretely. 1.4. Physical Assumptions. Assume that a rigid body of mass m executes a plane-parallel motion in a medium with quadratic resistance law and that a certain part of the exterior body surface is a plane plate under the medium streamline flow-around conditions. This means that the action of the medium on the plate reduces to the force S (applied at the point N ) whose line of action is orthogonal to the plate. Let the remaining part of the body surface be situated in a volume bounded by the flow surface that goes away from the plate boundary and is not subjected by the medium action. For example, similar conditions can arise after the body enters into the water. Assume that among the body motions, there exists a rectilinear translational drag regime. This is possible when the following two conditions hold: (1) the body velocity is orthogonal to the plate AB; (2) the perpendicular dropped from the body center of gravity C on the plate plane belongs to the line of action of the force S (Fig. 1).
Fig. 1: Rectilinear translational drag (unperturbed motion).
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1.4.1. Quasi-stationarity hypothesis and phase variables. Let us assign to the body the right coordinate system Dxyz whose axis z moves parallel to itself, and for simplicity, assume that the plane Dzx is the geometric symmetry plane of the body. This ensures the fulfillment of property (2) under the motion satisfying condition (1) (Fig. 2).
B x
C
D v
S
N A y
Fig. 2: Plane-parallel interaction of the body with the medium.
To construct the dynamical model, let us introduce the following phase coordinates: the value v = |v| of the velocity v of the point D (see Fig. 2), the angle α between the vector v and the axis x, and the algebraic value Ω of the projection of the body absolute angular velocity on the axis z. Assume that the value of the force S quadratically depends on v with nonnegative coefficient s1 (S = s1 v 2 ). One usually represents s1 in the form 1 s1 = ρP cx , 2 where cx is now the dimension-free coefficient of the frontal resistance (ρ is the medium density and P is the plate area). This coefficient depends on the angle of attack, the Strouhal number, and other quantities which are usually considered as parameters. In what follows, we also introduce the following additional phase variable of the “Strouhal type,” ΩD , ω= v where D is the characteristic plate transversal size (not to be confused with the point D). We restrict ourselves to the dependence of cx on the pair (α, ω) of variables, i.e., we assume that s1 (as well as yN ) is a function of the pair (α, ω) of dimension-free variables. Let us define (purely formally for now) the dependence of s1 and the ordinate yN of the point N on the phase coordinates (α, ω). The system of dynamical equations must admit a particular solution of the form α(t) ≡ 0,
ω(t) ≡ 0.
Therefore, we have the condition yN (0, 0) = 0 for the function yN (α, ω), and in the linear case, we need to assume that yN = D(kα − hω), where k and h are certain constants. Because the approximation is linear, we can ignore the dependence of s1 on α and ω. In what follows, to take account of the action direction of the force S, we introduce the following sign-alternating auxiliary function s(α, ω) = s1 (α, ω) sign cos α. 748
1.4.2. Key parameters. Therefore, the linearized model of the force medium action contains three parameters s, k, and h, which are determined by the plate form in the plan. As was already mentioned, the first of these parameters, the coefficient s, is dimensional. The parameters k and h are dimension-free because of the method of their introduction. Note that the quantities s and k can be found experimentally by using weight measurements in devices of the hydro- or aerodynamic tubes type. In [141,229], there is also information on the theoretical aspects of these quantities for separate plate forms (see also [97, 104, 106, 431, 437, 459]). This information allows us to assume that k > 0. As for the parameter h, even the very necessity of its introduction to the model is not a priori obvious. 1.5. Linearized Equations of Motion. The equations for motion of the center of mass in projections on the axes Dx and Dy of the related coordinate system and the equation of the kinetic moment variation with respect to the K¨ onig axis have the following form with accuracy up to terms linear in α and Ω (here, σ is the distance DC and I is the central moment of inertia of the body): v˙ = −
sv 2 , m
sv 2 α ˙ = 0, + vΩ − σ Ω m hDΩ 2 ˙ . I Ω = sDv kα − v
v α˙ −
(1) (2) (3)
Assuming that v = 0, introducing the natural parameter σ1 (v dt = D dσ1 ), which is usual for such systems, and using the change ω = DΩ/v (see above) of the variable Ω and the obvious differentiation formula d ( ) = v( ) D(˙) = v dσ1 we arrive at the system sv v = − D, (4) m Ds + sD3 kα, (5) Iω = ω(I − mD2 h) m sσD2 1 kσD h + sD + α, (6) α = −ω 1 + I m I in which the two latter equations are separated from the first, thus forming the independent second-order system (5), (6), and can be studied separately. Let us transform these equations as follows: exclude α from them introducing the angle of turn φ by the formula φ = ω; we obtain their linear integral in the form I σ σs Is hsD α−ω +φ 1+ + − = b = const. + kmD2 D m km2 D km Taking this into account, the equation for the angle of turn φ becomes kσs 2I Is hsD + φsD3 k + + 2 − = ksD3 b. Iφ + φ sD D2 h − Dσk − m m m D m It is easy to see that it has the form of the linear pendulum equation; under certain conditions, this pendulum executes oscillations near a certain position φ∗ determined by the value b of the linear integral. For h = 0, the coefficient of the so-called linear damping is negative, whereas the coefficient of the positional component is positive; this allows us to speak of the oscillatory instability of the solution φ = φ∗ . On the contrary, for a sufficiently large h, the solution φ = φ∗ can be made to be stable, and, moreover, it can lose its oscillatory character, since the coefficient of the positional component becomes negative. 749
But the main circumstance is that because this linear pendulum is multiparameter, the oscillatory stability of the solution φ = φ∗ is possible for certain finite values of h, since both mentioned coefficients can be positive in principle. 1.6. Experiment. To describe the results and concrete properties of the body motion, at the Institute of Mechanics of M. V. Lomonosov Moscow State University, V. A. Eroshin and V. M. Makarshin carried out experiments on registration of the motion of homogeneous circular cylinders in the water. Owing to the experiment, it becomes possible to find the dimension-free parameters k and h of the medium action on a rigid body; in particular, Sec. 3 is devoted to this. The experiment allows one to make several important conclusions. The first of them is as follows: the rectilinear stationary free body drag (in the water ) is unstable at least with respect to the angle of attack and the angular velocity. The second conclusion obtained from the carried out natural experiment is as follows: in modelling the medium action on the body, it is necessary to take account of the additional parameter characterizing the rotational derivative of the moment with respect to the body angular velocity. This parameter introduces the dissipation into the system. In our linear approximation, the considered damping moment depends linearly on the body velocity. In certain cases, the value of the damping moment coefficient under the body motion in the water was already estimated in [98, 99]. This estimate confirms the instability of the body rectilinear motion in the water. Purely formally, increasing the value of the damping coefficient, we can attain the stability of a motion, but it is difficult to ensure this stability in reality. The rigid body rectilinear motion is stable in certain media (for example in the clay), as the experiment shows [50–52]. Possibly, this stability is attained owing to the existence of a considerable damping from the medium in the system or for the existence of forces tangent to the plate. If the additional medium damping action on the body is of a purely dissipative character, then we can restrict ourselves to the region of positive damping moment parameter values, since a priori, its sign is not obvious. The value of this parameter is proportional to the transversal plate size, and to create the stability conditions of the motion considered, we also need to take the length of the moving body into account. Therefore, for sufficiently long bodies moving in the water, the contribution of the additional dissipation into the body orientation angle variations manifests itself only in a certain inessential decrease of the positive exponent of the exponential related to the instability of the motion. 1.7. Beginning of Nonlinear Analysis. The first conclusion made from the experiment forces us to consider the class of possible body motions for small angles of attack as a supporting class for studying the class of free body drag with finite angles of attack. For different bodies under motion in a medium and under certain conditions, the angles of attack can practically assume any value from the interval (0, π/2), and only for the angles close to π/2 is the so-called washing out of the lateral surface inevitable. Therefore, there arises the necessity of extending the functions yN and s to finite angles of attack, i.e., the pair of dynamical functions up to the interval (0, π/2). But, in fact, it is necessary to extend the dynamical functions to the whole numerical line; this is clear from the following arguments. Imagine an aircraft executing a plane-parallel motion over the water. In this case, the aircraft itself interacts with the water through a certain construction (for example, a keel) containing a plane surface part (plate) where the water flows around it when the aircraft moves over it. We can assume that the plane plate interacts with the water according to the streamline flow-around laws. Such an aircraft is similar to the well-known screencraft [207]. The plane-parallel motion of an aircraft over the water is ensured by stabilizing its transversal oscillations, in particular, for a screencraft, by the existence of the screen itself [83, 84, 207]. Under the motion of such an aircraft (or a screencraft) over the water, the mentioned plane plate can move in the water with practically arbitrary (real) angles of attack, including the case where it “cuts” the water, i.e., it moves with angles of attack close to π/2, as well as it assumes this value. Therefore, the dynamical functions of such a plate can be extended to any angles of attack. 750
1.7.1. Nonlinear equations. In order to pass to a more complete description of the free body motion, let us represent the dynamics equations ˙ =M mwc = F, I Ω (7) obtained previously in the linear form (see (1)–(3)) as follows: Fx , (8) m ˙ = 0, v˙ sin α + av ˙ cos α + Ωv cos α − σ Ω (9) DΩ . (10) I Ω˙ = yN (α, ω)Fx , ω = v As a rule, for various variants of the body motion considered below, the generalized force Fx is quadratic in velocities (v, Ω) and explicitly depends on the auxiliary sign-alternating function s(α, ω) (for example, Fx (α, v, Ω) = −s(α, ω)v 2 in the case of the body free drag). Therefore, the class of conceptual bodies and their conceptual motions defines a certain pair of dynamical functions s(α, ω), yN (α, ω) belonging to definite function classes. v˙ cos α − av ˙ sin α − Ωv sin α + σΩ2 =
1.7.2. Classes of dynamical functions. The first stage of the complete nonlinear study of the body motion in a medium under the quasi-stationarity conditions is the study of the corresponding dynamical systems in which the damping is not taken into account (in particular, h = 0 in the linear case). The account of the damping is the next labor-consuming stage of studying the problem, which is presented in this work in sufficient detail. To begin with, we consider the case where the pair of dynamical functions (yN , s) depends only on the angle of attack. In this case, to describe this pair of functions qualitatively, we use the experimental information on the streamline flow-around properties. The classes of dynamical functions to be introduced are sufficiently wide. They consist of sufficiently smooth 2π-periodic (yN (α) is odd and s(α) is even) functions satisfying the following conditions: (0) > 0 and y (π) < 0 (the function class {y } = Y ); yN (α) > 0 for α ∈ (0, π), and, moreover, yN N N s(α) > 0 for α ∈ (0, π/2), s(α) < 0 for α ∈ (π/2, π), and, moreover, s(0) > 0 and s (π/2) < 0 (the function class {s} = Σ). Both yN and s change sign upon replacement of α by α + π. Therefore, yN ∈ Y,
(11)
s ∈ Σ.
(12)
yN (α) = y0 (α) = A sin α ∈ Y,
(13)
In particular, the analytic functions s(α) = s0 (α) = B cos α ∈ Σ,
A, B > 0,
(14)
serve as typical representatives of the described classes and correspond to the medium interaction functions obtained by Chaplygin in studying the plane-parallel flow around of a plane plate of infinite length by a homogeneous medium flow. In what follows, there arises the product F (α) = yN (α)s(α) in the dynamical systems considered. It follows from the conditions listed above that F is a sufficiently smooth odd π-periodic function satisfying the following conditions: F (α) > 0 for α ∈ (0, π/2), F (0) > 0, and F (π/2) < 0 (the function class {F } = Φ). Therefore, F ∈ Φ. (15) In particular, the analytic function F = F0 (α) = AB sin α cos α ∈ Φ
(16)
is also a typical representative of the function class Φ arisen (and also corresponds to the Chaplygin case mentioned above). 751
Let us explain the necessity of a wide choice of function classes Y and Σ. A plane plate is a geometric section of the part of the body surface that interacts with the medium and is plane. The geometric form of such a plane domain can be arbitrary. Moreover, the chord lying in the plane of the domain can differently determine the plane of the body motion itself (in the case of the plane-parallel motion). The latter circumstances allow us to refer the dynamical functions arisen to definite classes. As was noted above, sufficiently weak conditions are imposed on these function classes, and, therefore, these classes are sufficiently wide. In advance, they include admissible concrete functions taken for each conceptual body and each conceptual motion. Therefore, to study the medium flow around a plate, we use the classes of dynamical systems defined by a pair of dynamical functions, which considerably complicates the global analysis. But certainly, it is not possible to associate a conceptual rigid body with its motion with each concrete pair of dynamical functions. Therefore, the study of this problem for sufficiently wide classes of dynamical functions allows us to speak of a relatively complete consideration of the problem of the body motion in a medium in the framework of these model assumptions under the quasi-stationarity conditions. 1.8. Directions Developed in the Work. Let us indicate the following directions developed in the work. The first two directions are traditional for analytical mechanics. (1) Elaboration of a qualitative methodology for studying nonlinear systems of dissipative character. Such a methodology allows us to answer a number of questions of nonlinear analysis, in particular, to the following question, which is principal for us: is it possible to find a pair of functions yn and s from certain classes such that in a finite neighborhood of the origin of the phase plane, there exist stable limit cycles of the separated system? (2) Search for possible integrable cases. It seems impossible to construct the general theory of studying systems of ordinary differential equations, even for systems of not very general form. Therefore, this work is not a next attempt in this direction. It only generalizes some qualitative studies in the dynamics of a rigid body interacting with a medium, which were initiated long ago. Therefore, we meet dynamical systems having very interesting properties. The third direction is characteristic for applied aerodynamics and is specific in the framework of this work. (3) Search for possible analogies between the clamped and free bodies dynamics. 1.8.1. Principal applied question of nonlinear analysis. The instability of the rectilinear translational drag allows us to pose the principal question of nonlinear analysis in studying a finite neighborhood of such a motion. Precisely, is it possible to find a pair of dynamical functions yN and s for describing the conceptual body motion such that in a finite neighborhood of this stationary motion, there exist stable limit cycles? One of the main results of this work is a partly negative answer to this question, precisely, for a quasistatic description of the interaction between the medium and the body, when the dynamical quantities yN and s depend only on the angle of attack, for any admissible pair of obtained dynamical functions yN (α) and s(α), in the whole range of finite angles of attack α ∈ (0, π/2), there are no auto-oscillations in the system considered. The mathematical aspect of this question is studied in Sec. 4 at a qualitative level. To attain a possible affirmative answer to the principal question of nonlinear analysis, in modelling the interaction of a body with a medium, we use an additional damping medium action, which gives a dissipation to the system. Therefore, as is shown in Sec. 10, in principle, under certain conditions, the appearance of stable auto-oscillations is possible in the framework of the model considered, but the search for a body exhibiting the necessary properties requires an additional experiment. This result is not only one of the main results of the present work, but it opens a new direction in the analytical study of the interaction of a body with a medium taking the medium damping action into account. 752
1.8.2. On variable dissipation in system. As is shown in Sec. 3, after certain simplifications, the general system (8)–(10) reduces to second-order pendulum systems in which there is a linear dissipative force with variable coefficient alternating the sign for different angles of attack. In this case, we therefore speak of the systems with the so-called variable dissipation, where the term “variable” mainly refers not to the value of the dissipation coefficient but to its sign. In the mean, during the period with respect to angle of attack, the dissipation can be positive, negative, or zero. In the latter case we speak about the system with variable dissipation with zero mean. 1.8.3. Mechanical and topological analogies. In what follows, it is necessary to note at once an important mechanical analogy arising on the basis of qualitative properties of the free body stationary motion in the medium flow and the pendulum equilibrium. Such an analogy allows us to extend the properties of pendulum nonlinear dynamical systems to the dynamical systems of a free body and obtain some topological analogies. For example, under condition (16), the angle of turn of the pendulum is completely equivalent to the angle of attack in the free body motion [233, 246–248, 276]. If condition (16) is violated (the group of conditions (13) and (14)), then the angle of attack of the free body and the angle of turn of the pendulum are trajectorially topologically equivalent (for such an equivalence, see Secs. 5, 6, and 8). 1.8.4. General character of symmetries in the system for plane and spatial dynamics. Moreover, under additional conditions such an equivalence also extends to the spatial case, which allows us to speak of the general character of symmetries which are in the system under the plane-parallel, as well as spatial motions (for the plane and spatial variants of the pendulum, see Secs. 6 and 8 and also [257, 258, 261, 265, 266, 268, 270–272, 274, 275, 277, 280, 281, 283–285, 289, 292, 293, 296, 297, 303, 306, 307, 310, 312, 314, 318, 321, 328, 329, 333, 336, 348, 349, 352, 355, 356, 362, 370, 371, 376, 378, 391, 393, 413, 414, 421]. Now we arrive at one more problem: the extension of the model for describing a free drag to the spatial case. This problem is the next labor-consuming stage of the study, and, moreover, the complexity of the natural experiment considerably increases in this case. Also, it should be noted that in studying the spatial free drag, we can obtain an analogous linear dynamical system, which allows us to use the methodology for finding the parameters of the medium action on the rigid body under the spatial motion (for such a methodology for the plane motion, see Sec. 3). In this case, the trajectory beams from the plane dynamics are the projections of analogous spatial beams. Note that, in reality, in performing every natural experiment, information on only one point is obtained. If a secondary repeated experiment is possible, then one can obtain information that is different from, although close to, the above information. Thus, the condensation of a large number of experimental points creates not only a complete picture of the real trajectory of the motion but certain difficulties in finding the real initial conditions, as well as dimension-free parameters of the medium action on the body. To overcome such difficulties, it seems expedient to obtain information on at least two points for each experimental trajectory that correspond to the same initial conditions of motion (for the experiment study, see Sec. 3) [251]. Since the principal question of nonlinear analysis was formulated above, after discussing the linearized problem, in Sec. 3, we form a number of nonlinear dynamical systems in the quasi-velocity space depending on two dynamical functions and describing various classes of the body motion in the medium under the quasi-stationarity conditions. A complete nonlinear analysis of such systems is performed in the next sections by the methods of qualitative theory known previously, as well as by new methods obtained exclusively for the arisen systems with variable dissipation. 1.8.5. General methodology of study. The assertions obtained in this work for variable dissipation system (see Sec. 3) are a continuation of the Poincar´e–Bendixson theory for systems on closed two-dimensional manifolds and the topological classification of such systems. The problems considered in this work stimulate the development of qualitative tools of studying, and, therefore, in a natural way, there arises a qualitative variable dissipation system theory. 753
In this case, we mainly focus on the topological classification of trajectory types and domains of trajectory location in the phase space. The analysis is performed analogously to the classical works [10–22, 45, 48, 53, 71, 72, 75, 88, 95, 115, 130, 131, 134, 138–140, 142, 143, 145, 148, 152, 153, 164–166, 169–176, 187, 188, 204, 205, 220–224, 226–228, 232, 252, 254, 256, 435, 438, 449, 452, 453, 455], as well as to the recent works [23–25,27–33,36–39,46,47,55,56,61,62,73,74,87,89–91,93,93,94,112–114,129,137,144,146,149–151, 156–163, 167, 186, 189, 191–198, 200–203, 212, 213, 216–218, 432–434, 442–448, 456, 460]. Let us dwell on the paper [24] (which reflects the full spectrum of problems of qualitative nature studied in the work) in more detail. This paper gives an overview of modern results related to the qualitative study of dynamical systems on closed two-dimensional manifolds, and, moreover, it mainly focuses on the problems of the complete topological study (including the classification). The scheme of a dynamical system on the sphere, the topological classification of Morse–Smale systems [432, 433] on two-dimensional manifolds, the so-called transitive [432] and singular systems on the torus, the homotopy class of the system semitrajectory rotation on closed orientable two-dimensional manifolds, and necessary and sufficient conditions for the topological equivalence of transitive systems on closed orientable two-dimensional manifolds are indirectly or directly related to the qualitative problems considered in this work (especially, in Sec. 4). This section is devoted to some problems of the qualitative theory of ordinary differential equations applied to concrete dynamical systems arising in the rigid body dynamics, as well as to arbitrary dynamical systems on low-dimensional smooth manifolds. Following Poincar´e [226,227], we improve some qualitative methods for finding key trajectories, i.e., the trajectories such that the global qualitative location of all other trajectories depends on the location and the topological type of these trajectories. Therefore, we can naturally pass to a complete qualitative study of the dynamical system considered in the whole phase space. In Sec. 4, we also obtain conditions for existence of the bifurcation birth of stable and unstable limit cycles for the systems describing the body motion in a resisting medium under the streamline flow-around. We find methods for finding any closed trajectories in the phase spaces of such systems and also present criteria for the absence of any such trajectories. We extend the Poincar´e topographical plane system theory and the comparison system theory to the spatial case. We study some elements of the theory of monotone vector fields on orientable surfaces. We suggest a sufficiently simple methodology for proving the Poisson stability of unclosed trajectories of dynamical systems. 1.9. Brief Contents of Other Sections of Work. In Sec. 5, we introduce the concepts of relative structural stability (relative roughness) and relative structural instability (relative nonroughness) of various degrees. The latter two properties are proved for the dynamical systems arising in the dynamics of a rigid body interacting with a resisting medium. In Sec. 6, we qualitatively study and integrate two model variants of the body plane-parallel motion in a resisting medium, which are described by dynamical systems with zero mean variable dissipation. One of the cases of motion presupposes the existence of a certain constraint in the system. Such a system is relatively structurally stable and topologically equivalent to a clamped pendulum placed in the over-running medium flow. We find a first integral of the system, which is a transcendental function of phase variables and expressed through elementary functions. The problems of plane rigid body dynamics described by systems with positive mean variable dissipation are qualitatively studied in Sec. 7. The problem that is most interesting from the applied viewpoint is the body free drag in a resisting medium. We obtain new families of topologically nonequivalent phase portraits. Almost all portraits of the family are (absolutely) structurally stable. In Sec. 8, we extend some results of the plane dynamics to the spatial case. In particular, we have integrated (in the Jacobi sense) the problem on the spatial motion of a dynamically symmetric clamped rigid body placed into an over-running medium flow. This system is topologically equivalent to the spatial rigid body motion in a resisting medium under which a certain constraint is imposed on the body. The spatial rigid body motion in a resisting medium in which the center of mass executes a rectilinear uniform motion is also a dynamical system with zero mean variable dissipation. Its qualitative study 754
allows us to present a spatial comparison system for studying many systems with nonzero mean variable dissipation. In Sec. 9, we obtain a new family of phase portraits in the problem of the spatial body free drag in a resisting medium. Almost all phase portraits of such a family are (absolutely) rough (structurally stable) in the Andronov–Pontryagin sense. In Sec. 10, we discuss some consequences of the introduced linear medium damping. In the problem of the body free drag in a medium, on the basis of nonlinear equations, we study the stability of the rectilinear translational drag subject to a linear damping moment. It is shown that in the framework of the model considered, in principle, there can arise the auto-oscillations corresponding to the limit cycles that are born from a weak focus (the well-known Andronov–Hopf bifurcation). The latter aspect is a possible affirmative answer to the principal question of nonlinear analysis. In the problem of the body motion in a medium under a certain constraint, we perform a complete nonlinear analysis of the dynamical systems in the quasi-velocity space. Such systems also exhibit the (absolute) rough property. We present a list of typical global phase portraits on the phase cylinder after surgeries of the phase portraits of analogous problems but without taking the damping into account. This section opens a new stage of research works in nonlinear analysis of body motion in a resisting medium under the quasi-stationarity conditions taking account of the medium damping. This section is the first stage of studying the body motion in a medium taking account of the medium damping moment. Such a moment introduces an additional dissipation (damping) into the system; as was already mentioned, this results in the rectilinear translation motion becoming stable in principle. Also, we show that for homogeneous circular cylinders moving in the water, the rectilinear translational drag is not stable for any dynamical and geometric parameters of such cylinders. Probably, this is related to the motion of the cylinders in the water, when the water damping is inessential, which does not allow us to speak of the stability of the rectilinear translational drag. However, for cylinders having a hole in their interior, the attainment of the above stability is possible under certain conditions. Therefore, under certain conditions, the account of the medium damping action on a rigid body leads to an affirmative answer to the principal question of nonlinear analysis: under the body motion in a medium with finite angles of attack, in principle, the appearance of stable auto-oscillations is possible. Moreover, for circular cylinders, the appearance of stable and unstable auto-oscillations is possible! All that said above allows us to estimate the results of the work as a whole as a new direction in analytical mechanics of a rigid body interacting with a medium. 2. Complete Integrability of Certain Classes of Nonconservative Systems As is shown in Sec. 1, the results of the presented work appeared owing to the study of the applied problem on the rigid body motion in a resisting medium [184, 185] where complete lists of transcendental first integrals expressed through a finite combination of elementary functions were obtained. This circumstance allows one to perform a complete analysis of phase trajectories and show those properties of them which exhibit the roughness and are preserved for systems of a more general form. The complete integrability of that systems is related to symmetries of latent type. Therefore, the study of sufficiently wide classes of dynamical systems having analogous latent symmetries is of interest. As is known, the concept of integrability is sufficiently broad and indefinite in general. In its constructing, it is necessary to take account of what it means (we mean a certain criterion according to which one concludes that the structure of trajectories of the dynamical system considered is especially “attractive and simple”), in which function class the first integrals are sought for, etc. (see [1, 27, 30, 48, 71, 82, 93, 95, 113, 119, 120, 123, 124, 126, 131, 133, 136, 153, 156, 162, 183, 187, 204, 227, 240, 243, 246, 255, 260, 277, 292, 294, 309, 315, 320, 323, 324, 331, 336, 342, 347, 357, 361, 369, 373–377, 379, 380, 396–400, 402–405, 409, 410, 416, 419, 420, 424, 435, 436, 447, 451, 460]). In this work, we follow the approach that takes the transcendental and, moreover, elementary functions as the function class for first integrals. Here the transcendence is understood not in the sense of elementary function theory (for example, trigonometrical functions), but in the sense that 755
these functions have essentially singular points (by the classification accepted in the theory of functions of one complex variable, when a function has essentially singular points). Moreover, we formally need to continue them to the complex domain. As a rule, such systems are strongly nonconservative (see [3, 6, 27, 71, 226, 246, 249, 309, 357, 365, 389, 394, 404, 405]). 2.1. Preliminaries. Of course, in the general case, it is quite difficult to construct a certain theory of integrating nonconservative systems (at least of lower dimension). But in a number of cases where the systems under study exhibit additional symmetries, we can succeed in finding their first integrals through finite combinations of elementary functions. In this work, we present examples from the dynamics of a rigid body interacting with a medium and from oscillation theory (see [405, 416]). In our previous works, we consider the class of problems from the rigid body dynamics in which the characteristic time of the body motion with respect to its center of mass is comparable with the characteristic time of motion of the center of mass itself. As was noted above, the complexity of solving such problems depends on many factors, including the character of the exterior force field. For example, in the case of conservative (gravity) force field, the motion of a body around its center of mass can be strongly chaotic (the classical problem of the heavy body motion around a fixed point) [153, 156]. In this case, it is not possible to construct a more or less general theory of integration; a natural possibility to proceed further is to impose some restrictions on the rigid body geometry and the necessity of existence of some groups of even latent symmetries for the force field considered. The presented work arises from the problem of the heavy body motion in a resisting medium, where the contact surface of the body with the medium is only a plane part of its exterior surface. In this case, the force field is constructed from the reasons for the medium action on the body in the streamline (or interrupted) flow around under the quasi-stationarity conditions. It turns out that the study of such a class of bodies reduces either to systems that are energy scattering (dissipative systems) or to systems that are energy supplying (the so-called antidissipative systems). Note that similar problems already appeared in applied aerodynamics in the studies of the N. E. Zhukovskii Central Aero- and Hydrodynamics Institute (see, e.g., [437]). Also, the classes of plane-parallel and spatial motions of rigid bodies interacting with a medium were previously considered; among them (depending on the number of degrees of freedom), we can highlight the following: the motion of bodies free in a medium that is immovable at infinity and bodies partially clamped and situated in the over-running medium flow [292, 294, 349]. One of such problems having the greatest applied significance was studied in detail; this is the problem of the body free drag in a resisting medium. Moreover, the problem of free body motion in the presence of a tracking force and also the problem of oscillation of a clamped pendulum in the over-running medium flow were considered. The problems considered previously stimulate the development of the qualitative tools for research, which essentially complement the qualitative theory of nonconservative systems with dissipation and antidissipation (i.e., systems with dissipative and dispersive forces). Therefore, we have studied the dynamical equations of motion arising in studying the plane and spatial dynamics of a rigid body interacting with a medium, which leads to a possible generalization of the method of study obtained previously to the general systems arising in the qualitative theory of ordinary differential equations, in dynamical system theory as well as in oscillation theory. Also, we have studied qualitatively nonlinear effects in the plane and spatial dynamics of a rigid body interacting with a medium; on the qualitative level, we have justified the necessity of introducing the concepts of relative roughness and relative nonroughness of various degrees (see [12, 24, 34, 35, 40, 69, 138, 140, 212, 216–218, 246, 266, 268, 269, 272, 275, 278, 286, 301, 322, 325, 326, 341, 351, 360, 384–387, 395, 415, 426, 432, 433, 452]). Performing the qualitative analysis, we previously did the following: (a) we elaborated the method of qualitative study of dissipative and antidissipative systems, which allows us to obtain the condition for birth bifurcation of stable and unstable auto-oscillations and 756
also the conditions for the absence of any singular trajectories. We have succeeded in extending the Poincar´e method of plane topographical systems and comparison systems to higher dimensions. We have obtained the sufficient Poisson stability conditions (conditions for everywhere density around itself) for certain classes on unclosed trajectories of dynamical systems (see [110, 111, 259, 262, 264, 273, 274, 282, 284, 285, 287, 288, 298–300, 304, 305, 316, 340, 353, 354, 358, 359, 368, 370, 405, 423, 427, 439]); (b) in the plane and spatial rigid body dynamics, we have discovered complete lists of first integrals of dissipative and antidissipative systems that are transcendental (in the sense of their singularity classifications) functions, which are expressed through elementary functions in a number of cases. We have introduced new definitions of properties of relative roughness and nonroughness properties that the integrated systems exhibit (see [4, 5, 125, 127, 243, 246, 250, 290, 294, 295, 309, 311, 313, 317, 320, 334, 335, 343, 344, 346, 350, 363, 372, 374, 376, 383, 417, 418]); (c) we have obtained multiparameter families of topologically nonequivalent phase portraits arising from the free drag problem. Almost all portraits of such families are (absolutely) rough (see [247, 249, 263, 267, 279, 291, 302, 308, 319, 327, 337, 338, 364, 388, 392, 422]); (d) we have discovered new qualitative analogies between the properties of the free body motion in a resisting medium immovable at infinity and clamped bodies situated in the over-running medium flow. The techniques developed for the study of these systems can also be used for solving problems of differential and topological diagnosis (see, e.g., [64–70,345,354,359,366,373,416,425,427–429]). Many results of this work were regularly reported at a number of workshops, including the workshop “Actual Problems of Geometry and Mechanics” named after professor V. V. Trofimov, which is headed by D. V. Georgievskii and M. V. Shamolin. Therefore, the work deals with certain aspects of mathematical modelling of the medium action on a rigid body under the quasi-stationarity conditions. These aspects form the initial concepts on problems of a qualitative nature arising further. 2.2. Variable Dissipation Dynamical Systems and Their General Properties. 2.2.1. General characteristic of variable dissipation dynamical systems. Generally speaking, the dynamics of a rigid body interacting with a medium is just a field, where there arise either dissipative systems or systems with the so-called antidissipation (energy supporting inside the system itself). Therefore, it becomes urgent to construct a methodology precisely for those classes of systems that arise in modelling body motion whose contact surface is a plane part, the simplest part of their exterior surface. Since in such a modelling, one uses the experimental information about the streamline flow-around properties, there arises a necessity of studying the class of dynamical systems that exhibit the (relative) structural stability property. Therefore, it is quite natural to introduce the definitions of relative roughness for such systems. In this case, many of the systems considered turned out to be (absolutely) rough in the Andronov–Pontryagin sense [19]. After certain simplifications, we can reduce the system of equations for the plane-parallel motion to the second-order pendulum systems in which there is a linear dissipative force with variable coefficient whose sign alternates for different values of the periodic phase variable in the system. In this case, we speak of systems with the so-called variable dissipation, where the term “variable” mostly refers not to the value of the dissipation coefficient but to a possible alternation of its sign (therefore, it is reasonable to use the term “sign-alternating”). In the mean during the period of the existing periodic coordinate, the dissipation can be either positive (“purely” dissipative systems), or negative (systems with dispersive forces), or zero. In the latter case, we speak of the systems with zero mean variable dissipation (such systems can be associated with the “almost” conservative systems). As was already noted, there are important mechanical analogies arising when comparing qualitative properties of the free body stationary motion and the pendulum equilibrium in the medium flow. Such analogies have a profound meaning, since they allow us to extend the properties of the pendulum nonlinear 757
dynamical systems to the free body dynamical systems. Both classes of systems belong to the class of the so-called pendulum dynamical systems with zero mean variable dissipation. Under additional conditions, we also extend the above equivalence to the case of spatial motion, which allows us to speak of the general character of symmetries in a system with zero mean variable dissipation under plane-parallel as well as spatial motions (for the plane and spatial variants of a pendulum in the medium flow, see [184, 246, 247, 292, 294, 320, 344, 401]). 2.2.2. Examples from dynamics. Below, we highlight the classes of essentially nonlinear systems of the second and third orders integrable in transcendental (in the sense of theory of functions of one complex variable) elementary functions. For example such systems are five-parametric dynamical systems, including the majority of systems that are previously studied in the dynamics of a rigid body interacting with a medium: α˙ = a sin α + bω + γ1 sin5 α + γ2 ω sin4 α + γ3 ω 2 sin3 α + γ4 ω 3 sin2 α + γ5 ω 4 sin α, ω˙ = c sin α cos α + dω cos α + γ1 ω sin4 α cos α + γ2 ω 2 sin3 α cos α + γ3 ω 3 sin2 α cos α + γ4 ω 4 sin α cos α + γ5 ω 5 cos α. In this connection, it is reasonable to introduce the definitions of relative structural stability (relative roughness) and relative structural instability (relative nonroughness) of various degrees. The latter two properties are proved for systems arising in the dynamics of a rigid body interacting with a medium, e.g., in [370, 405]. As is known, the (purely) dissipative dynamical systems (as well as (purely) antidissipative systems), which in our case can belong to the class of systems with zero mean variable dissipation, are as a rule structurally stable ((absolutely) rough), and, on the contrary, the systems with zero mean variable dissipation (which, as a rule, have additional symmetries) are either structurally unstable (nonrough) or only relatively structurally stable (relatively rough). It is difficult to prove the latter assertion in the general case. However, the introduction of the concept of relative roughness (and also relative nonroughness of various degrees) allows us to present the classes of concrete systems from the rigid body dynamics that exhibit the above properties. So, in [247], the authors qualitatively studied and integrated two model variants of the body plane-parallel motion in a resisting medium, which are described by dynamical systems with zero mean variable dissipation. Such cases of motion presuppose the existence of a certain nonintegrable constraint in the system considered (which is realized by a certain additional tracking force). For example a dynamical system of the form α˙ = Ω + β sin α,
Ω˙ = −β sin α cos α
is relatively structurally stable (relatively rough) and topologically equivalent to the system describing a clamped pendulum placed in the over-running medium flow [246] (its phase portrait is depicted in Fig. 3). We can find its first integral that is a transcendental (in the sense of theory of functions of one complex variable such that it has essentially singular points after continuing it into the complex domain) function of phase variables that is expressed through a finite combination of elementary functions. As is seen, the phase cylinder R2 {α, Ω} of quasi-velocities of the system considered exhibits an interesting topological structure of partition into trajectories. On the cylinder, there are two domains (whose closure is just the phase cylinder) filled with trajectories of absolutely different character. The first domain, called oscillatory or finitary (it is simply connected (see Fig. 3)), is entirely filled with trajectories of the following type. such trajectory starts at the repelling point (2πk, 0) Almost every and ends at the attracting point (2k + 1)π, 0 , k ∈ Z. Exceptions are the fixed points (πk, 0) and separatrices that either emanate from the repelling point (2πk, 0) and enter the saddles S2k and S2k+1 or 758
Fig. 3: Relatively rough dynamical system. emanate from the saddles S2k+1 and S2k+2 and enter the attracting points (2k + 1)π, 0 . Here π k Sk = − + πk, (−1) β . 2 The second domain, called rotational (it is two-connected (see Fig. 3)), is entirely filled with rotational motions similar to rotations on the phase plane of the mathematical pendulum. These phase trajectories envelope the phase cylinder and are periodic on it. Although the dynamical system considered is not conservative, in the rotational domain of its phase plane R2 {α, Ω} it admits the preservation of an invariant measure with variable density. This property characterizes the system considered as a system with zero mean variable dissipation. Key separatrices (for example, the separatrix emanating from the point (−π/2, β) and entering the point (3π/2, β)) are boundaries of domains in which the motion is of different character. So, in the oscillatory domain containing the repelling and attracting equilibrium points, almost all trajectories have attractors and repellers as limit sets. Hence there are no even absolutely continuous functions that are the density of an invariant measure in this domain. The matter is different for the domain entirely filled with rotational motions. As was shown above, there exists a smooth function that are the density of the invariant measure in the domain entirely filled with periodic trajectories not contractible to a point along the phase cylinder [246]. 2.2.3. First results. Some results of the dynamics of plane-parallel motion also extend to the spatial case; in this connection, the author has previously posed the spatial problem. In particular, we have found a complete list of integrals in the problem of the spatial motion (see, e.g., [292, 294, 320, 344, 401]) of a dynamically symmetric clamped rigid body placed in the over-running medium flow. This system with zero mean variable dissipation is topologically equivalent to the spatial motion of a rigid body in a resisting medium in which a nonintegrable constraint is imposed on the body (it is realized by a certain additional tracking force). The spatial motion of a rigid body in a resisting medium under which the center of mass executes the rectilinear uniform motion is also a dynamical system with zero mean variable dissipation. Its qualitative study allows us to present a convenient spatial comparison system for many systems with zero mean variable dissipation. Also, it should be noted that in [291,303,308,319,327,337,338,363,370,371,392,405] we have obtained a family of phase portraits in the problem of the spatial body free drag in a resisting medium. In these works, we develop the technique for studying a neighborhood of a singular equilibrium state, i.e., an 759
equilibrium state such that the right-hand sides of dynamical systems are extended to it only by continuity. For example, for small phase variables α and Ω, the right-hand side of the system has a singularity of the form 1/α. This difficulty is overcome by specifically constructing a Lyapunov function [340]. As a result of all this, we obtain a family of three-dimensional phase portraits analogous to the plane-parallel dynamics. 2.3. One of the Definitions of Systems with Zero Mean Variable Dissipation. We study a system of ordinary differential equations having a periodic phase coordinate. The systems under study have those symmetries for which in the mean during the period of the periodic coordinate, their phase volume is preserved. So, for example, the following pendulum system with smooth right-hand side V(α, ω) periodic in α of period T : α˙ = −ω + f (α),
ω˙ = g(α),
f (α + T ) = f (α), g(α + T ) = g(α),
preserves its phase area on the phase cylinder during the period T : T T T ∂ ∂ g(α) dα = f (α) dα = 0. div V(α, ω) dα = −ω + f (α) + ∂α ∂ω 0
0
0
The system considered is equivalent to the pendulum equation α ¨ − f (α)α˙ + g(α) = 0 in which the integral of the coefficient f (α) at the dissipative term α˙ vanishes in the mean during the period. It is seen that the system considered has those symmetries for which it becomes a system with zero mean variable dissipation in the sense of the following definition (see also [307,353,367,370,386,394,405]). Definition. Let us consider a smooth autonomous system of the (n + 1)th order and normal form defined on the cylinder Rn {x} × S1 {α mod 2π}, where α is a periodic coordinate of period T > 0. Denote by div(x, α) the divergence of the right-hand side (which is a function of all phase variables in general and is not identically equal to zero) of this system. Such a system is called a system with zero (nonzero) mean variable dissipation if the function T div(x, α) dα 0
is (is not) identically equal to zero. Moreover, in some cases (for example, when at certain points of the circle S1 {α mod 2π}, there arise singularities), this integral is understood in the principal value sense. It should be noted that it is rather difficult to give the general definition of a system with zero (nonzero) mean variable dissipation. The definition just presented uses the concept of divergence (as is known, the divergence of the right-hand side of a normal form system characterizes the variation of the phase volume in the phase space of the system). 2.4. Systems with Symmetries and Zero Mean Variable Dissipation. Let us consider systems of the form (the dot denotes derivative in time) α˙ = fα (ω, sin α, cos α), S1 {α
ω˙k = fk (ω, sin α, cos α), k = 1, . . . , n,
(17)
mod 2π} \ K × ω = (ω1 , . . . , ωn ), where the functions fλ (u1 , u2 , u3 ), defined on the set λ = α, 1, . . . , n, of three variables u1 , u2 , and u3 are as follows: fλ (−u1 , −u2 , u3 ) = −fλ (u1 , u2 , u3 ),
Rn {ω},
fα (u1 , u2 , −u3 ) = fα (u1 , u2 , u3 ),
fk (u1 , u2 , −u3 ) = −fk (u1 , u2 , u3 ).
The set K is either empty or consists of finitely many points of the circle S1 {α mod 2π}. The latter two variables u2 and u3 in the functions fλ (u1 , u2 , u3 ) depend on one parameter α, but they are distinguished in separate groups for the following reasons. First, not in the whole of their domain 760
are they uniquely expressed with respect to one another, and, second, the first of them is odd, whereas the second is an even function of α, which influences the symmetry of system (17) differently. To this system, we put in correspondence the nonautonomous system fk (ω, sin α, cos α) dωk = ; dα fα (ω, sin α, cos α) by the substitution τ = sin α, it reduces to the form (k = 1, . . . , n) fk (ω, τ, ϕk (τ )) dωk = , dτ fα (ω, τ, ϕα (τ ))
ϕλ (−τ ) = ϕλ (τ ), λ = α, 1, . . . , n.
In particular, the right-hand side of latter system can be algebraic (i.e., it can be a ratio of two polynomials); sometimes, this helps to search for its first integrals in explicit form. The following assertion imbeds the class of systems (17) in the class of dynamical systems with zero mean variable dissipation. The inverse embedding does not hold in general. Proposition 2.1. Systems of the form (17) are dynamical system with zero mean variable dissipation. This proposition is proved by using the symmetries of system (17) listed above. The converse assertion is not true in general since we can present a set of dynamical systems on a two-dimensional cylinder that are systems with zero mean variable dissipation, but they do not satisfy the properties listed above. In this work, we mainly consider the case where the functions fλ ω, τ, ϕk (τ ) (λ = α, 1, . . . , n) are polynomials in ω and τ . To begin with, let us consider a certain class of autonomous systems on the two-dimensional cylinder S1 {α mod 2π} × R1 {ω}. For example, to the following pendulum systems (arising in the dynamics of a rigid body interacting with a medium) with parameter β > 0 [246, 368, 430] (see also Fig. 3 as ω ↔ Ω and α ↔ −α): α˙ = −ω + β sin α, α˙ = −ω + β sin α cos α + βω sin α, 2
2
ω˙ = sin α cos α,
(18)
ω˙ = sin α cos α − βω sin α cos α + βω cos α 2
3
(19)
in the variables (ω, τ ), we put in correspondence the following equations with algebraic right-hand side, respectively: τ dω = , dτ −ω + βτ τ + βω[ω 2 − τ 2 ] dω = . dτ −ω + βτ + βτ [ω 2 − τ 2 ] In this case, systems (18) and (19) are dynamical systems with zero mean variable dissipation, which is easy to verify directly. Moreover, each of them has a first integral that is a transcendental (in the sense of the theory of functions of a complex variable) function that is expressed through a finite combination of elementary functions [246, 247]. For example, system (18) has a first integral of the following form (depending on the value of the constant β, three cases are possible that correspond to the existence of foci, nodes, or degenerate nodes in the phase portrait of the system):
2Ω + β sin α 2β 2 2 2 arctan = const; [Ω + βΩ sin α + sin α] × exp β − 4 < 0: sin α −β 2 + 4 −β 2 + 4 √ 2 √ 2 2Ω + β + β 2 − 4 sin α β −4−β × 2Ω + β − β 2 − 4 sin α β −4+β = const; β2 − 4 > 0 :
β sin α 2 = const. |2Ω + β sin α| × exp − β − 4 = 0: 2Ω + β sin α 761
Fig. 4: Relatively rough dynamical system with heteroclinic situation.
Fig. 5: Relatively nonrough dynamical system.
Fig. 6: Relatively rough dynamical system with homoclinic situation.
The phase portrait of system (19) can be of three different types depending on the values of the parameter β (Figs. 4, 5, and 6; ω ↔ Ω and α ↔ −α). In the expression of its first integral, three cases are also possible depending on the value of the constant β and corresponding to the existence of foci, nodes, and degenerate nodes in the phase portrait of the system. 762
Let us represent the parameter β as the product: β = σ 2 n20 ; after that, to system (19), we put in correspondence a differential equation of the form −n20 τ + σω[ω 2 − n20 τ 2 ] dω , = dτ ω + σn20 τ + στ [ω 2 − n20 τ 2 ] Introduce the following notation: C1 = 2 − σn0 ,
C2 = σn0 ,
τ = − sin α.
C3 = −2 − σn0 .
Performing a number of changes by the formulas ω − n0 τ = u1 ,
ω + n 0 τ = v1 ,
u 1 = v1 t 1 ,
v12 = p1 ,
where v1 = 0, we obtain the Bernoulli-type equation
dp1 2σ t 1 p1 = [C3 − C1 t21 ]. 2p1 C1 t1 + C2 + n0 dt1 By the known change p−1 = q1 for p1 = 0, we reduce the latter equation to the form q˙1 = a1 (t1 )q1 + a2 (t1 ), where
2(C1 t1 + C2 ) 4σt1 . , a2 (t1 ) = 2 C1 t1 − C3 n0 (C1 t21 − C3 ) (Here the dot denotes derivative in t1 .) The general solution of the linear homogeneous equation is represented in the form a1 (t1 ) =
q1 HOM (t1 ) = k(C1 t21 − C3 )Q(t1 ),
k = const,
where the function Q has the following form depending on the value of the constant C1 : t C1 = 0, e 1, 2 √ C2 arctan − C1 t1 C3 , C1 > 0, Q(t1 ) = e −C1 C3 C2 √ √ √ −C1 t1 + −C3 C1 C3 √ √ , C1 < 0. −C t − −C 1 1
3
To obtain the solution of the inhomogeneous equation, we assume that the quantity k is a function of t1 ; we find it by the quadrature 4σ t1 Q−1 (t1 ) dt1 . k(t1 ) = n0 (C1 t21 − C3 )2 Therefore, the transcendental first integral of system (19) becomes −1
Q
(t1 )q1 (C1 t21
−1
− C3 )
4σ − n0
t1 t0
Q−1 (τ1 )
τ1 dτ1 = C 0 , (C1 τ12 − C3 )2
C0
= const. where As is seen, the final form of the first integral depends on the sign of the constant C1 , and, therefore, three variants are possible. Let us examine each of them. First variant. C1 = 0. After an elementary calculation, we obtain an additional integral in the form u σ 2 u1 − v1 −2 +1 = const. e 1 v1 + 2 v1 Therefore, for C1 = 0, the transcendental first integral of system (19) is expressed through elementary functions. 763
Second variant. C1 > 0. The integration leads to the function C2 ζ C2 σ −2 √−C C3 1 √ e sin 2ζ + cos 2ζ + const, − 4n0 −C1 C3 where
ζ = arctan
−
C1 t1 . C3
As is seen, in the case C1 > 0, the additional first integral is expressed through elementary functions. Third variant. C1 < 0. By equivalent transformations, the integral transforms into 1−γ ζ ζ −γ ζ −1−γ σ 2 −3 + + const, C1 C2 n0 γ−1 γ γ+1 where C2 > 1, γ=√ C1 C3
√ √ −C1 t1 + −C3 √ ζ=√ . −C1 t1 − −C3
Therefore, in the case C1 < 0, the additional first integral is also expressed through elementary functions. Therefore, we study the connection between the following three properties, which are independent at first glance, but they are sufficiently harmonically combined on systems from the rigid body dynamics: (1) the distinguished class of systems (17) with the above symmetries; (2) the fact that this class of systems consists of systems with zero mean variable dissipation (in the variable α), which allows us to consider them as “almost” conservative systems; (3) in certain (although lower-dimensional) cases, these systems have first integrals, which are transcendental in general. Let us present one more important example of a higher-order system that has the properties just listed. To the system α˙ = −z2 + β sin α,
z˙2 = sin α cos α − z12
cos α , sin α
z˙1 = z1 z2
cos α , sin α
(20)
which is considered in the three-dimensional domain S1 {α mod 2π} \ {α = 0, α = π} × R2 {z1 , z2 } (such a system can be reduced to an equivalent system on the tangent bundle of the two-dimensional sphere) and describes the spatial motion of a rigid body in a resisting medium [405], we put in correspondence the following system with algebraic right-hand side: τ − z12 /τ dz2 = , dτ −z2 + βτ
dz1 z1 z2 /τ = . dτ −z2 + βτ
(21)
In this case, it is also seen that system (20) is a system with zero mean variable dissipation; in order to obtain a complete correspondence with the definition, it suffices to introduce the new phase variable z1∗ = ln |z1 |. Moreover, this system has two first integrals (i.e., the full list), which are transcendental functions and are expressed through a finite combination of elementary functions [294, 320]; as was mentioned above, this become possible after establishing its correspondence to the (nonautonomous in general) system of equations (21) with algebraic (polynomial) right-hand side. Therefore, systems from the rigid body dynamics presented above not only enter the class of systems (17) and have mean zero variable dissipation, but they have a full list of transcendental first integrals expressed through a finite combination of elementary functions. In this case, the integration of systems (18) and (19) reduces to the integration of the corresponding equations with algebraic right-hand sides. 764
As was noted, to seek the first integrals of the systems considered, it is better to reduce systems of the form (17) to systems with polynomial right-hand sides; the possibility of integrating the initial system in elementary functions depends on their form. Therefore, we proceed as follows: we seek sufficient conditions for integrability in elementary functions of systems of equations with polynomial right-hand sides studying systems of the most general form in this case. 2.5. Systems on S1 {α mod 2π} × R1 {ω} or Two-Dimensional Cylinder. Let us consider the possibility of completely integrating (in elementary functions) a system of the form x˙ = ax + by + f1 x3 + f2 x2 y + f3 xy 2 + f4 y 3 , y˙ = cx + dy + g1 x3 + g2 x2 y + g3 xy 2 + g4 y 3 on the plane R2 {x, y}. Moreover, for definiteness, we assume that the roots of the characteristic equation of the λ-matrix a−λ b c d−λ are real. In this case, we can assume that b = 0. In the case where there exist complex conjugated roots of this equation, we can assume that ad − bc = 1. Applying the substitution y = tx, which is characteristic for homogeneous systems, we arrive at the integration of the identity [at + bt2 + f1 tx2 + f2 t2 x2 + f3 t3 x2 + f4 t4 x2 − c − dt − g1 x2 − g2 tx2 − g3 t2 x2 − g4 t3 x2 ] dx + [ax + btx + f1 x3 + f2 tx3 + f3 t2 x3 + f4 t3 x3 ] dt = 0. It is seen that eight parameters characterize the nonlinearity. To integrate the latter identity in elementary functions, it suffices to impose the following five independent relations: g1 = 0,
f1 = g2 = β1 ,
f2 = g3 = β2 ,
f3 = g4 = β3 ,
f4 = 0.
Proposition 2.2. The seven-parameter family of systems of equations x˙ = ax + by + β1 x3 + β2 x2 y + β3 xy 2 , y˙ = cx + dy + β1 x2 y + β2 xy 2 + β3 y 3 on the plane R2 {x, y} has a (transcendental in general ) first integral expressed through elementary functions. A scheme of proof. After simple transformations, we arrive at the necessity to integrate the Bernoulli equation dx + [a + bt]x + [β1 + β2 t + β3 t2 ]x3 = 0; [at + bt2 − c − dt] dt by the change x−3 = u, it reduces to the linear inhomogeneous equation at + bt2 − c − dt du + [a + bt]u = −β1 − β2 t − β3 t2 . − 2 dt Since the primitive of the function a + bt at + bt2 − c − dt is expressed through elementary functions, it follows that the general solution of the homogeneous part of the latter equation is also expressed through elementary functions. 765
If, without loss of generality, we assume that b = 0, then a particular solution of the latter inhomogeneous equation is expressed through a finite combination of elementary functions. Remark. Also, we can seek the first integral of the system from the equation 2 · x x x = (a − d) + b − c . y y y Corollary 2.1. For any parameters a, b, c, d, β1 , β2 , and β3 , systems of the form α˙ = a sin α + bω + β1 sin3 α + β2 ω sin2 α + β3 ω 2 sin α, ω˙ = c sin α cos α + dω cos α + β1 ω sin2 α cos α + β2 ω 2 sin α cos α + β3 ω 3 cos α have a transcendental first integral expressed through elementary functions. In particular, system (18) is obtained from the latter system for a = β,
b = −1,
c = 1,
d = β1 = β2 = β3 = 0,
and system (19) for a = β,
b = −1,
c = 1,
d = −β,
β1 = −β,
β2 = 0,
β3 = β.
Let us generalize the previous arguments. We consider the possibility of completely integrating (in elementary functions) systems of a more general form. Now, the nonlinearity is characterized by an arbitrary homogeneous form of odd degree 2n − 1. Then the following assertion, more general than Proposition 2.2, holds. Proposition 2.3. The (2n + 3)-parameter family of systems of equations x˙ = ax + by + δ1 x2n−1 + δ2 x2n−2 y + · · · + δ2n−2 x2 y 2n−3 + δ2n−1 xy 2n−2 , y˙ = cx + dy + δ1 x2n−2 y + δ2 x2n−3 y 2 + · · · + δ2n−2 xy 2n−2 + δ2n−1 y 2n−1
(22)
on the plane R2 {x, y} has a (transcendental in general ) first integral expressed through elementary functions. The family of equations (22) indeed depends on 2n − 1 + 4 independent parameters since in this case the general nonlinearity of odd degree is characterized by 4n parameters on which 2n + 1 conditions are imposed (four more parameters sit in the linear part). Corollary 2.2. For any parameters a, b, c, d, δ1 , . . . , δ2n−1 , systems of the form α˙ = a sin α + bω + δ1 sin2n−1 α + δ2 ω sin2n−2 α + · · · + δ2n−1 ω 2n−2 sin α, ω˙ = c sin α cos α + dω cos α + δ1 ω sin2n−2 α cos α + δ2 ω 2 sin2n−3 α cos α + · · · + δ2n−1 ω 2n−1 cos α have a transcendental first integral expressed through elementary functions. Let us make the following important remark. As was mentioned, systems (18) and (19) are relatively rough, but if the symmetries are violated in these systems (for example, if we add additional terms to their right-hand sides), which were introduced for systems of a more general form (17), then the number of topologically different phase portraits can strongly change. So, for example, the following systems that are a “perturbation” of a system of the form (19) become dynamical systems with nonzero mean variable dissipation and have infinitely many topologically nonequivalent phase portraits (see, e.g., Figs. 7–16, the portraits of a multiparameter family of systems without limit cycles, and Figs. 17–24, the portraits of a multiparameter family of systems, when under certain relations for the parameters of the systems, there exist simple and complicated limit cycles) [289,302]. 766
Fig. 7: Family of portraits without limit cycles.
Fig. 8: Family of portraits without limit cycles.
Fig. 9: Family of portraits without limit cycles.
Fig. 10: Family of portraits without limit cycles.
767
Fig. 11: Family of portraits without limit cycles.
Fig. 12: Family of portraits without limit cycles.
Fig. 13: Family of portraits without limit cycles.
Fig. 14: Family of portraits without limit cycles.
768
Fig. 15: Family of portraits without limit cycles.
Fig. 16: Family of portraits without limit cycles.
Fig. 17: Family of portraits with limit cycles.
Fig. 18: Family of portraits with limit cycles.
769
Fig. 19: Family of portraits with limit cycles.
Fig. 20: Family of portraits with limit cycles.
Fig. 21: Family of portraits with limit cycles.
Fig. 22: Family of portraits with limit cycles.
770
Fig. 23: Family of portraits with limit cycles.
Fig. 24: Family of portraits with limit cycles.
2.6. Systems on S1 {α mod 2π} \ {α = 0, α = π} × R2 {z1 , z2 } or Tangent Bundle of TwoDimensional Sphere. Let us study a system of the form (20) that reduces to system (21) and also the system α˙ = −z2 + β(z12 + z22 ) sin α + β sin α cos2 α, z˙2 = sin α cos α + βz2 (z12 + z22 ) cos α − βz2 sin2 α cos α − z12 z˙1 = βz1 (z12 + z22 ) cos α − βz1 sin2 α cos α + z1 z2
cos α , sin α
cos α , sin α
which also arises in the spatial dynamics of a rigid body interacting with a medium and corresponds to the following system with algebraic right-hand side: τ + βz2 (z12 + z22 ) − βz2 τ 2 − z12 /τ dz2 , = dτ −z2 + βτ (z12 + z22 ) + βτ (1 − τ 2 ) βz1 (z12 + z22 ) − βz1 τ 2 + z1 z2 /τ dz1 . = dτ −z2 + βτ (z12 + z22 ) + βτ (1 − τ 2 )
(23)
In a similar way, we pass to the homogeneous coordinates uk , k = 1, 2, according to the formulas zk = uk τ . System (21) reduces to the system τ
τ − u21 τ du2 + u2 = , dτ −u2 τ + βτ
τ
du1 u1 u2 τ + u1 = , dτ −u2 τ + βτ 771
which, in turn, corresponds to the equation 1 − βu2 + u22 − u21 du2 = . du1 2u1 u2 − βu1
(24)
This equation is integrated in elementary functions since the identity 1 − βu2 + u22 + du1 = 0 d u1 is integrated and has the following first integral in the coordinates (τ, z1 , z2 ) (cf. [346, 362]): z12 + z22 − βz2 τ + τ 2 = const. z1 τ After the reduction, system (23) corresponds to the system τ
τ + βu2 τ 3 (u21 + u22 ) − βu2 τ 3 − u21 τ du2 , + u2 = dτ −u2 τ + βτ 3 (u21 + u22 ) + βτ (1 − τ 2 )
τ
βu1 τ 3 (u21 + u22 ) − βu1 τ 3 + u1 u2 τ du1 , + u1 = dτ −u2 τ + βτ 3 (u21 + u22 ) + βτ (1 − τ 2 )
which also reduces to (24). A system of the form (20) is equivalent to the following system on the tangent bundle of the two-dimensional sphere T∗ S2 (in this case, β → −β): sin θ = 0, θ¨ + β θ˙ cos θ + sin θ cos θ − ψ˙ 2 cos θ
1 + cos2 θ ψ¨ + β ψ˙ cos θ + θ˙ψ˙ = 0. sin θ cos θ sin α Its phase portrait is depicted in Fig. 25 (for α = θ and z1 = ψ˙ cos α ).
Fig. 25: Relatively rough phase portrait in a three-dimensional domain.
Let us pose the following question: What is possibility of integrating the system ax + by + cz + c1 z 2 /x + c2 zy/x + c3 y 2 /x dz = , dx dx + ey + f z gx + hy + iz + i1 z 2 /x + i2 zy/x + i3 y 2 /x dy = dx dx + ey + f z 772
(25)
of a more general form in elementary functions and three-dimensional phase domains, which includes systems (21) and (23) and has a singularity of the form 1/x? As before, introducing the substitutions y = ux and z = vx, we obtain that system (25) reduces to the system ax + bux + cvx + c1 v 2 x + c2 vux + c3 u2 x dv +v = , dx dx + eux + f vx gx + hux + ivx + i1 v 2 x + i2 vux + i3 u2 x du +u= ; x dx dx + eux + f vx x
to the latter system, we put in correspondence the following equation with algebraic right-hand side: a + bu + cv + c1 v 2 + c2 vu + c3 u2 − v[d + eu + f v] dv = . du g + hu + iv + i1 v 2 + i2 vu + i3 u2 − u[d + eu + f v] The integration of the latter equation reduces to the integration of the following equation in total differentials: [g + hu + iv + i1 v 2 + i2 vu + i3 u2 − du − eu2 − f uv] dv = [a + bu + cv + c1 v 2 + c2 vu + c3 u2 − dv − euv − f v 2 ] du. (26) In general, we have the 15-parameter family of equations of the form (26). To integrate the latter identity as a homogeneous equation in elementary functions, it suffices to impose seven relations g = 0,
i3 = e,
i1 = 0,
i = 0,
c2 = e,
c = h,
2c1 = i2 + f.
(27)
Introduce eight parameters β1 , . . . , β8 g = 0,
h = β1 ,
f = β5 ,
a = β6 ,
i1 = 0, b = β7 ,
i = 0,
i2 = β2 ,
c = β1 ,
i3 = β3 , d = β4 , e = β3 , β2 + β5 , c2 = β3 , c3 = β8 c1 = 2
and consider them as independent parameters. Therefore, under the group of conditions (27), Eq. (26) reduces to the form β6 + β7 u + (β1 − β4 )v + (β2 − β5 )v 2 /2 + β8 u2 dv = ; du (β1 − β4 )u + (β2 − β5 )vu
(28)
after that, the equation under study is integrated in elementary functions. Indeed, integrating identity (26), we obtain the relation
(β2 − β5 )v 2 β6 (β1 − β4 )v +d +d − d[β7 ln |u|] − d[β8 u] = 0; d u 2u u in the coordinates (x, y, z), it allows us to obtain the first integral in the form y (β2 − β5 )z 2 /2 − β8 y 2 + (β1 − β4 )zx + β6 x2 − β7 ln = const. yx x
(29)
Therefore, we can make a conclusion on the integrability in elementary functions of the following, in general, nonconservative third-order system depending on eight parameters: β6 x + β7 y + β1 z + (β2 − β5 )z 2 /2x + β3 zy/x + β8 y 2 /x dz = , dx β4 x + β3 y + β5 z β1 y + β2 zy/x + β3 y 2 /x dy = . dx β4 x + β3 y + β5 z 773
Corollary 2.3. On the set S1 {α mod 2π} \ {α = 0, α = π} × R2 {z1 , z2 }, the third-order system α˙ = β4 sin α + β3 z1 + β5 z2 , z˙2 = β6 sin α cos α + β7 z1 cos α + β1 z2 cos α +
β2 + β5 2 cos α cos α cos α z2 + β3 z1 z2 + β8 z12 , 2 sin α sin α sin α
(30) cos α cos α z˙1 = β1 z1 cos α + β2 z1 z2 + β3 z12 sin α sin α depending on eight parameters has a (transcendental in general) first integral expressed through elementary functions. In particular, for β1 = β3 = β7 = 0, β2 = β6 = 1, β5 = β8 = −1, and β4 = β, system (30) reduces to system (20) To find an additional first integral of the nonautonomous system (25), we use the first integral (29) just found, which is expressed through a finite combination of elementary functions. 2.7. Systems from Dynamics of a Four-Dimensional Body Interacting with a Medium. 2.7.1. Enlarging the dimensionality in rigid body dynamics. In [246,247], the authors showed the complete integrability of the plane (i.e., the body is situated in the Euclidean space E2 ) problem on the rigid body motion in a resisting medium under the conditions of streamline flow around in the case where the system of dynamical equations has one first integral that is a transcendental (having essentially singular points in the sense of the theory functions of a complex variable) function of the quasi-velocities. It was assumed that the interaction of the medium with a body is concentrated on a part of the body surface that has the form of a (one-dimensional ) plate. Later [294, 297, 320], the plane problem was generalized to the spatial (three-dimensional ) case (i.e., the motion is executed in the Euclidean space E3 ), and, moreover, the system of dynamical equations has a complete tuple of first integrals. Here it was assumed that the interaction of a medium with a body is concentrated on a part of the body surface that has the form of a plane (two-dimensional ) disk. The structure of the dynamical equations of motion is often preserved under the extension of dynamical properties to cases of higher dimension. For example, at present (see [30, 56–59, 116–123, 206]), the theory of a four-dimensional (and moreover, n-dimensional) body motion has been developed. The authors of recent works (of the S. P. Novikov and O. Ya. Bogoyavlenskii school) succeeded in showing the Hamiltonian property of the equations of many-dimensional rigid body motion around a fixed point. The present section is devoted to the study of the motion of a four-dimensional body interacting with a medium according to the “streamline flow-around” laws and presents the results of studying this problem for the first time. We assume that the whole interaction of a (four-dimensional) rigid body with a medium is concentrated on a part of the (smooth three-dimensional ) body surface that has the form of a (three-dimensional ) ball. In this case, the angular velocity tensor is six-dimensional, whereas the velocity of the center of mass is four-dimensional. 2.7.2. Statement of the problem and equations on algebra so(4). Let a four-dimensional body move in a resisting medium that fills a four-dimensional domain of the Euclidean space E4 , and let the whole interaction of the medium with the body be concentrated on a part of the (smooth three-dimensional) body surface that has the form of the three-dimensional disk D3 . The distance from the point N of the resistance force application to the center D of the disk is a function of only one parameter, the angle of attack α, which is measured between the velocity v of the point D and the middle perpendicular to the disk dropped from the center of mass C of the body in the four-dimensional space (see [329, 331, 332, 339, 342, 347–349, 355, 356, 370, 375, 379, 380, 389, 397, 399, 403, 406, 411, 418]). 774
The resistance force is orthogonal to the disk D3 in the four-dimensional space, and its value has the form S = s1 (α)v 2 , where s1 is the nonnegative resistance coefficient. Let us relate the coordinate system Dx1 x2 x3 x4 to the body whose axis Dx1 coincides with the axis CD, and the axes Dx2 , Dx3 , and Dx4 lie in the disk hyperplane. If in the system Dx1 x2 x3 x4 , the inertia operator has the diagonal form diag{I1 , I2 , I3 , I4 }, where Ω is the angular velocity tensor of the rigid body, Ω ∈ so(4), then the part of the four-dimensional rigid body motion equations corresponding to the algebra so(4) has the form (see [444, 447]) ˙ + ΛΩ ˙ + [Ω, ΩΛ + ΛΩ] = M, ΩΛ
(31)
where
λ1 =
−I1 + I2 + I3 + I4 , 2
Λ = diag{λ1 , λ2 , λ3 , λ4 }, I1 − I2 + I3 + I4 I1 + I2 − I3 + I4 λ2 = , λ3 = , 2 2
λ4 =
I1 + I2 + I3 − I4 , 2
M is the exterior force moment acting on the body in R4 and projected on the natural coordinates in the algebra so(4), and [. . .] is the commutator in so(4). It is convenient to represent the matrix Ω ∈ so(4) in the natural coordinates: 0 −ω6 ω5 −ω3 ω6 0 −ω4 ω2 , (32) −ω5 ω4 0 −ω1 ω3 −ω2 ω1 0 where ω1 , ω2 , ω3 , ω4 , ω5 , and ω6 are components of the angular velocity tensor in projections on the natural coordinates in the algebra so(4). It is convenient to represent the resistance coefficient s1 in the form s1 (α) = s(α) sign cos α. If (0, x2N , x3N , x4N ) are the coordinates of the point N in the system Dx1 x2 x3 x4 , and {−S, 0, 0, 0} are the coordinates of the resistance force vector in the same system, then in calculating the moment of the resistance force, it is necessary to construct the mapping R4 × R4 −→ so(4)
(33)
that transforms a pair of vectors from R4 into a certain element of the algebra so(4). In projections on coordinates in the group so(4), the moment of the resistance force has the form (0, 0, x4N S, 0, −x3N S, x2N S) ∈ R6 ∼ = M ∈ so(4),
(34)
Here it is necessary to take into account that if (v, α, β1 , β2 ) are the spherical velocity coordinates of the center D of the three-dimensional disk in R4 , then x2N = R(α) cos β1 ,
x3N = R(α) sin β1 cos β2 ,
x4N = R(α) sin β1 sin β2 . 775
Taking all this into account, we can obtain the following equations of motion in the considered nonconservative resistance force field: (λ4 + λ3 )ω˙1 + (λ3 − λ4 )(ω3 ω5 + ω2 ω4 ) = 0,
(35)
(λ2 + λ4 )ω˙2 + (λ2 − λ4 )(ω3 ω6 − ω1 ω4 ) = 0,
(36)
(λ4 + λ1 )ω˙3 + (λ4 − λ1 )(ω2 ω6 + ω1 ω5 ) = x4N S,
(37)
(λ3 + λ2 )ω˙4 + (λ2 − λ3 )(ω5 ω6 + ω1 ω2 ) = 0,
(38)
(λ1 + λ3 )ω˙5 + (λ3 − λ1 )(ω4 ω6 − ω1 ω3 ) = −x3N S,
(39)
(λ1 + λ2 )ω˙6 + (λ1 − λ2 )(ω4 ω5 + ω2 ω3 ) = x2N S.
(40)
2.7.3. Dynamics on R4 . By analogy with the three-dimensional case, we can deduce formulas similar to the Euler and Rivals formulas: the velocities and accelerations of any two points A and B of the four-dimensional rigid body are related as follows in any coordinate system: vB = vA + ΩAB,
wB = wA + Ω2 AB + EAB,
(41)
˙ ∈ so(4). The matrix E is called the acceleration matrix (corresponding to the where Ω ∈ so(4), E = Ω tensor of the acceleration) of a four-dimensional rigid body. Using formulas (41), we can obtain the equations of motion for the center of mass of the four-dimensional rigid body in R4 . 2.7.4. Motion in a resisting medium under the action of a nonintegrable constraint. Let us consider the class of body motions under which the following condition holds all the time: v = |v| = const.
(42)
In this case, we assume that a certain (tracking) traction force acts on the body ensuring the fulfillment of condition (42) and being the reaction of the nonintegrable constraint (cf. the two- and three-dimensional cases [246, 320]). We can attain the fulfillment of condition (42) by a definite choice of the traction value along the line CD (see [247]). 2.7.5. Case of a dynamically-symmetric rigid body. By analogy with the three-dimensional case, let the following relation hold: (43) I2 = I3 = I4 . In such a case, there exist the following three cyclic first integrals of Eqs. (35)–(40): ω1 = ω10 ,
ω2 = ω20 ,
ω4 = ω40 .
For simplicity, let us consider the motions at their zero levels ω10 = ω20 = ω40 = 0.
(44) To describe the body motion, we use the pair of dynamical functions R(α), s(α) ; the information about them is of a qualitative nature. By analogy with the cases of sufficiently lower dimensions, without loss of generality [85, 86, 246], we can assume that R(α) = A sin α, A > 0,
s(α) = B cos α, B > 0.
(45)
Then the equations corresponding to so(4) take the form (here, n20 = AB/2I2 ) ω˙3 = n20 v 2 sin α cos α sin β1 sin β2 , ω˙5 = ω˙6 = 776
−n20 v 2 sin α cos α sin β1 cos β2 , n20 v 2 sin α cos α cos β1 .
(46) (47) (48)
If we introduce the natural change of angular velocities by the formulas z1 = ω3 cos β2 + ω5 sin β2 ,
(49)
z2 = −ω3 sin β2 cos β1 + ω5 cos β2 cos β1 + ω6 sin β1 ,
(50)
z3 = ω3 sin β2 sin β1 − ω5 cos β2 sin β1 + ω6 cos β1 ,
(51)
then the complete system of dynamical equations of motion takes the following symmetric form on the direct product so(4) × R4 (after the account of four Eqs. (42) and (44)): α˙ = −z3 + σn20 v sin α, z˙3 = n20 v 2 sin α cos α − (z12 + z22 )
cos α , sin α
cos α cos α cos β1 + z12 , sin α sin α sin β1 cos α cos α cos β1 , − z1 z2 z˙1 = z1 z3 sin α sin α sin β1 cos α , β˙1 = z2 sin α cos α β˙2 = −z1 . sin α sin β1 z˙2 = z2 z3
(52)
(53)
The sixth-order system (52), (53) has an independent fifth-order system (52). For the complete integration of this system, in general, we need to know five independent first integrals. However, after the change of variables z2 (54) z1 , z2 −→ z = z12 + z22 , z∗ = , z1 the system (52), (53) reduces to the form α˙ = −z3 + σn20 v sin α, z˙3 = n20 v 2 sin α cos α − z 2
cos α , sin α
cos α , z˙ = zz3 sin α cos α cos β1 z˙∗ = 1 + z∗2 z , sin α sin β1 zz∗ cos α , β˙1 = 1 + z∗2 sin α cos α β˙2 = −Z1 (z, z∗ ) . sin α sin β1
(55)
(56)
(57)
It is seen that the fifth-order system (52) falls into independent systems of lower order: system (55) is of the third order, and system (56) (of course, after the change of the independent variable) is of the second order. Therefore, for the complete integrability of the system (55)–(57), it suffices to find two independent first integrals of system (55), one integral of system (56), and an additional first integral, which “ties” Eq. (57). System (55) arises in the three-dimensional rigid body dynamics [320]. It has two transcendental integrals: z 2 + z32 − σn20 vz3 sin α + n20 v 2 sin2 α = C1 = const, z z sinz α 3 , , sin α = C2 = const. G sin α sin α
(58) (59) 777
System (56) has a first integral of the form 1 + z∗2 = C3 = const, sin β1
(60)
and, in turn, the additional first integral “attaching” Eq. (57) has the form cos β1 ± 2 = sin{C3 (β2 + C4 )}, C3 − 1
C4 = const.
(61)
2.7.6. A new direction. Previously, only those motions of a four-dimensional body were considered for which M ≡ 0 (or the force field is conservative). This work opens a new direction developed by the author in studying the equations of the rigid body motion on the direct product so(4) × R4 (M is not identically equal to zero) in a nonconservative force field. Sometimes, the methodology of integrating the dynamical systems considered can be also extended to the space so(n) × Rn of an arbitrary dynamically-symmetric n-dimensional rigid body. Therefore, the dynamical systems considered in this work refer to systems with zero mean variable dissipation with respect to the existing periodic coordinate. Moreover, such systems often obey a complete list of first integrals expressed through elementary functions. The method for reducing the initial systems of equations with right-hand sides containing polynomials in trigonometric functions to systems with polynomial right-hand sides allows one to seek first integrals (or to prove their absence) also for systems of a more general form but not only for those which have the above symmetries. 3. Class of Generating Dynamical Systems on Lower-Dimensional Manifolds: Once More about Resisting Medium Action on a Rigid Body Despite the facts of the previous section, which are exactly of mathematical character, the author considers that it is necessary to include the following material concerning applications to this work. This material probably makes the reader confident that the systems of ordinary differential equations (including the generating ones) studied here are taken from a model problem of rigid body motion in a fluid, which is of great importance. Therefore, in the mathematical model of the rigid body drag in a resisting medium under streamline flow-around and quasi-stationarity conditions, the translational drag regime (see Fig. 1) is as a rule unstable. This allows us to elaborate a sufficiently simple methodology for experimentally finding the parameters of the medium action on the body considered. We present an example of the use of this methodology for a body of cylindrical form; precisely, we consider the problem of a circular cylinder entering into water. This study comprises the part devoted to studying the linear interaction model. As is shown in Sec. 1, in what follows, we restrict ourselves mainly to extending the model to the nonlinear case described by the system (8)–(10). Therefore, at the end of this section, we obtain a number of nonlinear dynamical systems describing several variants of the body motion. Such systems are variable dissipation systems. 3.1. Preliminaries. To begin with, in this section, we study the linearized equations for the body drag in a medium. Recall that in [184, 185], the problem of stability of the free body rectilinear drag was already examined. The study is based on the linearized equations. It is shown that the instability of such a motion is sometimes related to the swing of the body angular oscillations. However, it is difficult to ensure the uniform body motion experimentally. Therefore, we consider the body drag problem to be more convenient for the experimental verification of the swing effect. The preparatory studies [105] allow one to detect it and show that for a better quantitative coincidence of the calculational and experimental trajectories, it is necessary to complement the quasi-static model by adding one more parameter, the moment rotational derivative in the body angular velocity. This parameter usually introduces a dissipation into the system considered, but it turns out to be insufficient to reject 778
the swing effect here. This study allows us to elaborate a sufficiently simple and effective methodology for finding the unknown model parameters. Along with the angle ϕ determining the body orientation on the plane, it is quite natural to introduce the ordinate of the point D (see Figs. 1 and 2) in a fixed coordinate system, i.e., the lateral displacement a considered as a function of the natural dimension-free parameter τ . Therefore, the pair of functions ϕ(τ ), a(τ ) uniquely defines the body position on the plane in the linear approximation. Under the condition a(0) = 0, we can define the lateral displacement a(τ ) by the formula τ a(τ ) =
α(q) + ϕ(q) dq.
(62)
0
Of course, it is difficult to think that the unstable motion has any applied significance. However, such a motion can be used for methodological purposes in order to find the unknown parameters, in particular, the dimension-free parameters k and h of the medium action on the body considered. In the next section, we pass to the methodology for finding these parameters. 3.2. Methodology for Finding Unknown Dimension-Free Parameters of Medium Action on a Body. As was mentioned above, the linearized model of the resisting medium force action on the body considered contains three parameters s, k, and h, which are determined by the plate form in the plan. The first of these parameters, the resistance coefficient s, is dimensional. The latter two parameters k and h are dimensionless by the method of their introduction. At the Institute of Mechanics of M. V. Lomonosov State University (cf. [109, 147, 242, 251, 461]), Eroshin and Makarshin carried out experiments on the registration of the motion of two homogeneous circular cylinders in water with the following parameters: m = 178 g, 2σ = 35 mm,
D = 30 mm
(steel model),
(63)
m = 178 g, 2σ = 60 mm,
D = 30 mm,
(titanium model)
(64)
with velocities v ∼ = 150–250 m/sec (cx = 0.82 for both cylindrical models). The registration was performed by taking a photograph of the cylinder at some points of the trajectory on the interval 0 < τ < 45. From the photo, they took the longitudinal displacement τi of the cylinder point D along the line L1 (situated in the fixed space) of the unperturbed motion, the transversal displacement ai = a(τi ) of the point D, and the angle ϕi = ϕ(τi ) of the cylinder axis orientation with respect to the line L1 . Some experimental results are depicted by points in Figs. 26–29 (for the steel model). Such pictures can also be presented for the titanium model. For both models, the nominal initial conditions were chosen as follows: ϕ0 = 0, ω = ω0 , and α0 = ω0 σ/D. The experimental dependence ϕ(τi ) can be interpreted as oscillations near a certain negative value with increasing amplitude, which can serve as a manifestation of the translational motion instability described above. Moreover, the dispersion of experimental points, especially with respect to the lateral displacement, allows one not only to assume measurement errors but errors in the initial condition realization. Therefore, for the processing, the so-called “covering beam” method was used. The essence of this method, as applied to the problem considered, is that the experimental data depicted in Figs. 26–29 by points is covered by beams of curves that are solutions of the corresponding differential equation for various initial conditions. In this case, the initial conditions are chosen such that it is possible to cover the whole experimental point net satisfactory. As in [101] (cf. [96]), for the parameter k, the estimate k = 0.1 is accepted, whereas for the initial conditions in the angles of attack and deviation, it is taken near 0. For each of the models, three groups of trajectories were constructed for various initial conditions for h = 0, h = 0.1, and h = 0.2. In this case, the angular velocity was chosen in the limits 10−2 –10−3 . 779
Fig. 26: Variation of deviation angle.
Fig. 27: Variation of lateral displacement.
Fig. 28: Variation of deviation angle.
Fig. 29: Variation of lateral displacement.
Each such group contains trajectory beams for various initial conditions. So, for example, for the steel model, it is possible to show the integral trajectory beams corresponding to the initial conditions ϕ0 = 00 ; −0.50 ; −10 ; −1.50 ;
ω0 = −0.02;
α0 = 00 .
Also, the pictures corresponding to the initial conditions ϕ0 = 00 ;
α0 = 00 ;
ω0 = −0.01; −0.015; −0.02
are of interest, and the initial conditions are ϕ0 = 00 ;
ω0 = −0.017;
α0 = −10 ; −0.50 ; 00 ; 0.50 .
In a similar way, all this is also performed for the titanium model. The images in Figs. 27 and 29 (for the steel model) show that all beams satisfactorily “cover” the experimental points a(τi ). At the same time, the extent of correspondence for ϕ(τi ) (Figs. 26 and 28) is different for different values of h, which allows us to accept the value hexperimental = 0.1 as a preliminary estimate. Also, it should be noted that the dispersion of the initial conditions that allows us to “cover” the experimental points by a “net” of solutions of the corresponding differential equation in the angles of attack and deviation, as well as in the initial velocity, lies in the limits of the order as the initial conditions for the so-called “canonical” trajectories [109, 251]. In conclusion, we note that it is already shown on the basis of linear equations of motion that the unstable character of the rectilinear translational drag can be used for constructing a methodology for finding the parameters of the body interaction with a medium. But the body angular oscillation swing effect leads to the necessity to take into account the nonlinear terms in the complete system of equations. 780
Therefore, in the next section, we present the nonlinear dynamical systems describing various variants of the body motion. 3.3. Nonlinear Dynamical Systems Describing Various Variants of Body Motion in a Medium. The nonlinear model of the interaction of a body with a medium was studied above. As was shown above, we restrict ourselves mainly to the nonlinear model of the interaction described by system (8)–(10) under conditions (11), (13) (or (15)). As was already mentioned, one of the goals of the present work is the study of a finite neighborhood of the rectilinear translational drag in a part of the variables (α, Ω) and the finding of a pair (yN , s) of dynamical functions that “detects” possible auto-oscillations in the system considered. Since the instability of the rectilinear translational motion holds in a part of the variables (α, Ω), we consider the possibilities of separating an independent subsystem on the plane from the general system (8)–(10). For this purpose, we consider the classes of motions that allow us to find the character of changing the quantity v and thus to perform the study of the pair (α, Ω) of variables as independent variables. The quantity v must satisfy a certain condition, so that to provide it, we need to impose the controlling traction T on the system, which acts, e.g., along the line CD (see Fig. 2). Let us show sufficient conditions for separating an independent second-order subsystem from the general system (8)–(10). Assume that the value of the traction is represented in the form T = τ1 (α)v 2 + τ2 (α)vΩ + τ3 (α)Ω2 , where τi (α), i = 1, 2, 3, are sufficiently smooth functions. Then the system (8)–(10) is the Euler homogeneous system in a part of the quasi-velocities (Ω, v) of homogeneity degree 2. Indeed, after the change of the independent variable (time t) according to the formula d¯ q = v dt, under which the Jacobian v of the change vanishes nowhere, from (8)–(10) we obtain the new system ϕ = ω,
(65)
v = vΨ(α, ω), α = −ω + ω =
(66)
σ s(α) − τ¯(α, ω) F (α) cos α + σω 2 sin α + sin α, I m
1 F (α) − ωΨ(α, ω), I
(67) (68)
where Ψ(α, ω) =
τ¯(α, ω) − s(α) σ F (α) sin α − σω 2 cos α + cos α, I m τ¯(α, ω) = τ1 (α) + τ2 (α)ω + τ3 (α)ω 2 .
In the third-order system (66)–(68), there arises the independent second-order subsystem (67), (68) which can be considered separately on its two-dimensional phase cylinder. 3.3.1. Body motion under a certain constraint. In what follows, let us consider the plane-parallel body motion under the condition that a nonintegrable servo-constraint of the following form is imposed on the system: (69) v = v0 = const. In this case, Eq. (8) (in which the value of the longitudinal component of the total force acting on the body is represented in the form Fx = T − s(α)v 2 ) holds identically. In the phase space, Eq. (69) defines the class of constraint level surfaces Φ0 = v = const. In this case, the function T is the control (servo-constraint reaction modulus) that supports the motion of the system under condition (69). 781
Under condition (69) in the system (8)–(10), there arises an independent subsystem of the form αv˙ cos α + Ωv cos α − σ Ω˙ = 0,
˙ = F (α)v 2 IΩ
(70)
in which the parameter v is added to the constant parameters indicated above (see [110, 111, 183, 233, 245–248, 257–259, 263, 266, 268, 275]). As is seen from (70), on the manifold π (71) Λi , Λi = (α, Ω) ∈ R2 : α = + πi O= 2 i∈Z
it is impossible to resolve the system uniquely with respect to α. ˙ Therefore, formally, on the manifold O, we have the uniqueness theorem violation. From this, it follows that outside the manifold O and only there, system (70) is equivalent to the system F (α) ˙ = A2 F (α), , Ω (72) cos α σv v2 , A2 = ; A1 = I I we study this system under condition (15) (on the manifold O, we define the system being studied by continuity). In particular, if condition (16) holds, then system (72) takes the form of the analytic system α˙ = −Ω + A1
α˙ = −Ω + A1 sin α,
˙ = A2 sin α cos α, Ω v 2 AB
(73)
σvAB , A2 = . I I The uniqueness theorem violation for system (70) on the manifold O means the following: through almost every point of O, a nonsingular trajectory of system (70) passes and intersects O at a right angle and also there exists a phase trajectory that entirely coincides with the above point at all instants of time. But physically, these are different trajectories since different values of the traction T correspond to them. Let us show this. To satisfy the constraint of the form (69), it is necessary to choose the value of reaction T in the form mσ F (α) tan α . (74) Tv (α, Ω) = σmΩ2 + v 2 s(α) − I (On manifold O, we define it by continuity.) Let F (α) = L. lim α→π/2 cos α Note that |L| < +∞ if and only if π F < +∞. 2 For α = π/2, the desired traction value is found from the relation π mσ 2 , Ω = σmΩ2 − Lv , Tv 2 I where the value of Ω is arbitrary. On the other hand, to support the rotation around a certain point W using the traction T, we need to choose the value of the traction in the form π mv 2 ,Ω = , (75) Tv 2 R0 where R0 is the distance CW . A1 =
782
In general, relations (74) and (75) define different values of T = Tv for almost all points of the manifold O; this proves the remark. 3.3.2. Motion with a constant velocity of the center of mass. The dynamical system (8)–(10) being studied can also describe the class of motions under which the velocity of the center of mass is constant as a vector [245, 247, 248, 257, 258, 260]. In particular, the latter circumstance is attained by the choice of the control traction from the condition T = −S. (76) Condition (76) is also a sufficient condition for the order reduction of the system (8)–(10); this system can be transformed into the form ϕ = ω,
(77)
v = vΨ(α, ω), σ α = −ω + F (α) cos α + σω 2 sin α, I 1 ω = F (α) − ωΨ(α, ω), I
(78) (79) (80)
where
σ F (α) sin α − σω 2 cos α. I Condition (77) (as well as (65)) is a new kinematic relation, whereas the system (78)–(80) (as well as (66)–(68)) is a new complete system considered in the dynamical space of quasi-velocities. As before, the function F entering the right-hand side of the system (78)–(80) satisfies condition (15). In particular, if condition (16) holds, then the system (78)–(80) takes the form Ψ(α, ω) =
v = vΨ(α, ω), AB α = −ω + σ sin α cos α + σω 2 sin α, I AB sin α cos α − ωΨ(α, ω), ω = I where Ψ(α, ω) = σ
(81) (82) (83)
AB sin2 α cos α − σω 2 cos α. I
3.3.3. Rigid body free drag. Let us return to the consideration of the body free drag (see [107–109, 184, 185, 249, 251, 257, 258, 261, 270, 279, 289]). We have T ≡ 0.
(84)
Condition (84) (as well as (74), (75), and (76)) is sufficient for the order reduction of the system (8)–(10), which can be transformed to the form ϕ = ω, v = vΨ(α, ω), α = −ω + ω =
σ s(α) F (α) cos α + σω 2 sin α + sin α, I m
1 F (α) − ωΨ(α, ω), I
(85) (86) (87)
where
s(α) σ F(α) sin α − cos α. I m Equations (86) and (87) compose a closed second-order subsystem. In this case, we assume that the functions F and s entering the right-hand side of the system (85)–(87) satisfy conditions (15) and (13). Ψ(α, ω) = −σω 2 cos α +
783
In particular, if conditions (16) and (14) hold, then the system (85)–(87) takes the form v = vΨ(α, ω), σ B α = −ω + AB sin α cos2 α + σω 2 sin α + sin α cos α, I m 1 ω = AB sin α cos α − ωΨ(α, ω), I
(88) (89) (90)
where Ψ(α, ω) = −σω 2 cos α +
σ B AB sin2 α cos α − cos2 α. I m
3.3.4. Nonlinear reduced oscillator. For example, let us consider systems of the form (72) on the plane. Let F (α) g(α) = cos α (on the manifold O, we define it by continuity). By (15), the function g is sufficiently smooth. After the change y2 = −Ω + A1 g(α), y1 = α, regular in R2 {α, Ω}, the phase trajectories of system (72) are equivalent to the phase trajectories of the system y˙1 = y2 , y˙2 = −A2 g(y1 ) cos y1 + A1 y2 g (y1 ), which describes a nonlinear pendulum with variable dissipation. The motion is executed under the action of the following two forces: the potential force A2 g(y1 ) cos y1 and the linear-in-velocity force y2 whose coefficient A1 g (y1 ) alternates its sign in general. In those bands on the phase cylinder on which the coefficient A1 g (y1 ) is strictly positive, we have the medium energy pumping. In those bands where the linear-in-velocity force coefficient y2 is strictly negative the force acting on the body scatters the energy forcing the pendulum to put the brake on its motion. The latter remark shows that we are dealing with a dissipative system with variable dissipation (more precisely, with a system with sign-alternating dissipation). The phase plane R2 {α, Ω} is composed of the union of alternating domains in which there is a dissipation of only one sign. 3.3.5. More about dissipative systems and systems with zero and nonzero mean variable dissipation. The systems (72); (73); (79), (80); (82), (83) are systems with zero mean variable dissipation. It is not easy to give a general and strict definition of such systems. In the present work, we define such systems as follows. A system on the phase cylinder is a system with zero mean variable dissipation if the dissipation is equal to zero in the mean over the cylinder. For example, for system (73), the phase cylinder is the closure of the union of the bands π π Π = (α, Ω) ∈ R2 : − < α < 2 2 and π π Π = (α, Ω) ∈ R2 : < α < 3 2 2 in each of which the dissipation of the system has a strictly definite sign: in the band Π, it is negative (i.e., there is energy pumping), and in the band Π , it is positive (i.e., energy scattering occurs). The specific feature of energy pumping or scattering in this case is the sign of the divergence of the system vector field. Indeed, the above divergence for system (73) is equal to A1 cos α and is positive in the band Π, 784
which explains the phase volume “expansion” by the phase flow, and in the band Π , it is negative, which explains its “diminishing” (see Sec. 6). In the mean over the cylinder, the divergence is defined as 2π cos α dα = 0.
A1 0
We show below that the listed systems have additional symmetries, and, therefore, their phase cylinders are unions of alternating bands in each of which there exists only a one-sign dissipation. On the contrary, the systems (86), (87) and (89), (90) have no property described above, despite the fact that on their phase cylinders, there indeed exist dissipations of different signs. In the mean over the phase cylinder, the dissipation is strictly positive for the above systems (see Sec. 7), and, therefore, these systems are systems with nonzero mean variable dissipation. In the next sections, we pass to constructing the qualitative methodology for studying the obtained dynamical systems. Analogous systems obtained from the spatial dynamics are studied by the analogous method of the many-dimensional qualitative theory. The study of spatial dynamics systems (because of its complexity) is presented in Secs. 8 and 9. 4. Some Problems of Qualitative Theory of Ordinary Differential Equations in Variable Dissipation System Dynamics In this section, we consider some problems of qualitative theory of ordinary differential equations; the study of dissipative dynamical systems, as well as variable dissipation systems considered below, which, in particular, arise in the dynamics of a rigid body interacting with a medium and oscillation theory, depends on solutions of these problems. We consider such problems as the bifurcation of the birth of a limit cycle from a weak focus; the existence problems of the so-called monotone limit cycles; the existence of closed trajectories contractible to a point along two-dimensional surface or trajectories not contractible to a point along the phase cylinder; the qualitative problems of Poincar´e topographical system theory and the more general comparison systems; existence and uniqueness problems of trajectories having infinitely distant points as limit sets for systems on the plane; elements of qualitative theory of monotone vector fields and also existence problems of families of long-period and Poisson stable trajectories. In conclusion, we study the possibility of extending the Poincar´e two-dimensional topographical system theory and the comparison system theory to the many-dimensional case. 4.1. Remarks on Bifurcation of Birth of Poincar´ e–Andronov–Hopf Cycle. 4.1.1. General theorems on birth of a limit cycle. The goal of this section is to propose a sufficiently simple, although not exhaustive presentation of the phenomenon, which is usually called the bifurcation of birth of limit cycle from a weak focus, and also its applications to problems of dynamics of a rigid body interacting with a resisting medium. Historically, the problem goes back to the works [226, 227] of H. Poincar´e (1892). Also, this theme was discussed by A. A. Andronov [10–20] starting from 1930. The main works of Hopf devoted to this problem appeared in 1942. Although the term “Poincar´e–Andronov–Hopf bifurcation” (sometimes even the name of Friedrichs is included here) is more precise, the term “Hopf bifurcation” is more widespread in the western literature. The reason for this is that the most essential contribution of Hopf is the generalization of this result from the two-dimensional case to the case of higher dimensions. The cycle birth bifurcation is the appearance of periodic orbits (“auto-oscillations”) from a stable fixed point under passage of the parameter through a critical value. Although the possibility of applying the Hopf bifurcation theory to nonlinear parabolic partial differential equations is proved at present, we restrict ourselves to the application of this theory to ordinary differential equations on the plane. As was mentioned above, the bifurcation of birth of a stable cycle from a weak asymptotically stable equilibrium state usually refers to the above bifurcation [10–20]. 785
Theorem 4.1 ([74, 226, 227, 405]). On the plane, let us consider an autonomous system x1 = X1 (x1 , x2 , µ), x2 = X2 (x1 , x2 , µ)
(91)
defined by sufficiently smooth functions X1 (x1 , x2 , µ) and X2 (x1 , x2 , µ) depending on the parameter µ. For any µ, let the system have a fixed point at the origin. If λi (µ) (i = 1, 2) are eigenvalues of the linearized system at the origin and they are complex-conjugate (and also purely imaginary for µ = µ0 ), then let d Re λi (µ) > 0 (< 0). dµ µ=µ0
Moreover, assume that for µ = µ0 , the origin is asymptotically stable. Then (1) µ = µ0 is a bifurcation point of the system; (2) there exists µ1 < µ0 (µ1 > µ0 ) such that for any µ ∈ (µ1 , µ0 ) (µ ∈ (µ0 , µ1 )), the origin is a stable focus; (3) there exists µ2 > µ0 (µ2 < µ0 ) such that for any µ ∈ (µ0 , µ2 ) (µ ∈ (µ2 , µ0 )), the origin is an unstable focus surrounded by a stable limit cycle whose size grows when µ increases (decreases) from µ0 up to µ2 as |µ − µ0 |. The proof of this (and also a more general) theorem can be found in [74]. The invariant manifold method is the main technique used in the proof. The essence of the proof is the application of the implicit function theorem. Using Theorem 4.1, we can formulate an analogous theorem on the unstable limit cycle birth. Theorem 4.2. Let system (91) have a fixed point at the origin for any µ. If λi (µ) (i = 1, 2) are eigenvalues of the linearized system and they are complex conjugate (and also purely imaginary for µ = µ0 ), then let d Re λi (µ) < 0 (> 0). dµ µ=µ0
Moreover, assume that for µ = µ0 , there exists a neighborhood of the origin such that any nontrivial trajectory leaves it in a finite time. Then (1) µ = µ0 is a bifurcation point of the system; (2) there exists µ1 < µ0 (µ1 > µ0 ) such that for any µ ∈ (µ1 , µ0 ) (µ ∈ (µ0 , µ1 )), the origin is an unstable focus; (3) there exists µ2 > µ0 (µ2 < µ0 ) such that for any µ ∈ (µ0 , µ2 ) (µ ∈ (µ2 , µ0 )), the origin is a stable focus surrounded by an unstable limit cycle whose size grows when µ increases (decreases) from µ0 up to µ2 as |µ − µ0 |. Theorem 4.2 can be proved either by the method used in proving Theorem 4.1 or by reducing system (91) to a new system via the change t ⇒ −t of the independent variable and the change µ ⇒ −µ of the parameter in which there arises the birth of a stable limit cycle in accordance with Theorem 4.1. Remark. The theorems presented above are essentially related to one-parameter families of differential equations. Therefore, in general, there are no uniform estimates for parameters under which there exists the cycle birth bifurcation. Therefore, if the system of equations considered depends on m parameters, then in general, the cycle birth is related to the change of only one parameter for other m − 1 parameters being fixed. Remark. The main reason for the existence of the above bifurcation is the existence of a nonrough equilibrium state (but only a focus, which is an attractor or a repeller). We can present examples in which all conditions of the theorem hold except for the following one: in the linear, as well as nonlinear case, the equilibrium state is a center. The latter condition is a key condition in this problem. It contradicts the rough limit cycle birth (cf. [74, 405]). 786
4.1.2. Limit cycle birth in the problem of rigid body motion in a medium under a nonholonomic constraint. Let us consider systems F(α) , Ω = A2 F(α) − hΩ, (92) α = −Ω + A1 cos α of the form (72) with additional summand on the set π π . Π = (α, Ω) ∈ R2 : − < α < 2 2 For h = 0, the pendulum system (92) coincides with (72). For h = 0, the system contains a linear damping moment [263] (see Sec. 10). Lemma 4.1. Let
F (0) >0, F(−α) = −F(α), F(ε0 ) > 0 for all ε0 ∈ 0, π2 , A21 F (0) − A2 < 0; (1) A1 > 0, (2) F ∈ C 2 − π2 , π2 , |F(α)| < F (0)|α| cos α (|F(α)| > F (0)|α| cos α) in a certain neighborhood of the origin (or F (0) + 3F (0) < 0 (F (0) + 3F (0) > 0)); π π d F (α) (3) dα cos α ≥ 0 for all α ∈ − 2 , 2 .
Then (1) for h ≤ 0, there are no closed curves of trajectories in the strip Π; (2) the point h = h0 = A1 F (0) is a bifurcation point of the system; (3) there exits h1 < h0 such that for any h ∈ (h1 , h0 ), the point (0, 0) is an unstable focus surrounded by a unique stable limit cycle whose size grows when h decreases from h0 down to h1 (for any h ∈ (h1 , h0 ), the point (0, 0) is an unstable focus); (4) there exists h2 > h0 such that for any h ∈ (h0 , h2 ), the point (0, 0) is a stable focus (for any h ∈ (h0 , h2 ), the point (0, 0) is a stable focus surrounded by a unique unstable limit cycle whose size grows when h grows from h0 up to h2 ). This lemma is proved by the improved Bendixson method [45] and also by constructing an auxiliary Lyapunov function. Remark. As is shown below, Condition (c) of Lemma 4.1 is often extra. Corollary 4.1. The function F = F0 = AB sin α cos α, AB > 0 (see (16)) satisfies the condition of Lemma 4.1. In this case, there arises a stable limit cycle in the strip Π. Lemma 4.2. In the strip Π ,
π π Π = (α, Ω) ∈ R2 : < α < 3 , 2 2
under the conditions of Lemma 4.1 and also under the condition F(α + π) = −F(α), let us consider systems of the form (92). Then (1) for h ≥ 0, there are no closed curves of trajectories in the strip Π ; (2) the point h = h0 = −A1 F (0) is a bifurcation point of the system; (3) there exists h1 > h0 such that for any h ∈ (h0 , h1 ), the point (π, 0) is a stable focus surrounded by a unique unstable limit cycle whose size grows when h grows from h0 up to h1 (for any h ∈ (h0 , h1 ), the point (π, 0) is a stable focus); (4) there exists h2 < h0 such that for any h ∈ (h2 , h0 ), the point (π, 0) is an unstable focus (for any h ∈ (h2 , h0 ), the point (π, 0) is an unstable focus surrounded by a unique limit cycle whose size grows when h decreases from h0 down to h2 ). Lemma 4.2 follows from Lemma 4.1 and the Chetaev instability theorem. 787
4.1.3. Limit cycle birth in the problem of rigid body free drag in a resisting medium. Let us consider systems of the form (86), (87) in the strip Π under conditions (15) and (13) unless otherwise specified. If F (0) , B = s(0), n20 = I then we introduce two dimensionless parameters µ1 = 2
B > 0, mn0
µ2 = σn0 > 0.
Let µ = µ1 − µ2 . In what follows, we obtain sufficient conditions for existence of a weak attractor and those for existence of a weak repeller, which allow us to find the cycle birth bifurcation. Lemma 4.3. Let us consider the system (86), (87) in the strip Π . Let π π π π F ∈ C3 − , ∩ Φ, s ∈ C 2 − , ∩ Σ, 2 2 2 2 and also let σ s2 σn0 < 2, In = − f3 + 4 − 4σn20 , f1 = F (0), f3 = F (0), I m Then
s2 = s (0).
(1) if In < 0, then (a) for µ1 = µ2 , there is a bifurcation of the system; ¯1 < 0 such that for any µ ¯ = µ1 − µ02 ∈ (¯ µ1 , 0), the point (π, 0) (b) fix µ02 < 2. Then there exists µ ¯ = µ1 − µ02 ∈ (0, µ ¯2 ), the point (π, 0) is a stable focus; there exists µ ¯2 > 0 such that for any µ is an unstable focus surrounded by a stable limit cycle whose size grows when µ ¯ grows from 0 up to µ ¯2 ; (2) if In > 0, then (a) for µ1 = µ2 , there is a bifurcation of the system; ¯1 > 0 such that for any µ ¯ = µ1 −µ02 ∈ (0, µ ¯1 ), the point (π, 0) is (b) fix µ02 < 2. Then there exists µ an unstable focus; there exists µ ¯2 < 0 such that for any µ ¯ = µ1 − µ02 ∈ (¯ µ2 , 0), the point (π, 0) is a stable focus surrounded by an unstable limit cycle whose size grows when µ ¯ diminishes from 0 down to µ ¯2 . In Lemma 4.3, there are no unform estimates for both parameters µ1 and µ2 . Corollary 4.2. Let the conditions (16), (14), and σn0 < 2 hold. Then B = σn0 , there is a bifurcation of the system; (1) for 2 mn 0 ¯1 < 0 such that for any µ ¯ = µ1 − µ02 ∈ (¯ µ1 , 0), the point (2) fix σn0 = µ02 < 2. Then there exists µ (π, 0) is a stable focus; there exists µ ¯2 > 0 such that for any µ ¯ = µ1 − µ02 ∈ (0, µ ¯2 ), the point (π, 0) is an unstable focus surrounded by a stable limit cycle whose size grows when µ ¯ grows from 0 up to µ ¯2 .
Remark. The chain of relations µ1 = µ2 , In = 0,. . . defines a certain set in the infinite-dimensional space of Taylor series expansion coefficients of the functions F and s. Because of the fact that for µ1 = µ2 or In = 0, we succeed in finding the character of trajectories near the point (π, 0), the quantities µ = µ1 − µ2 and In are functions of the first two coefficients of the first return function. Indeed, µ = 0 and In = 0 if and only if either the linear coefficient or the coefficient at the cube of the first return function expansion vanishes [226]. 788
4.2. On Closed Curves of Trajectories Contractible to a Point along Phase Surface. In this section, we study the problems of existence of closed curves composed of the trajectories for systems of differential equations on two-dimensional surfaces. The phase curves considered contract to a phase surface point. Therefore, the desired closed phase trajectories compose a part of the trivial component of the fundamental group of a given surface. The importance of finding such a class of closed curves composed of the phase trajectories is that for any smooth two-dimensional phase manifold, there exist curves contractible to a point along it. Although the latter curves can be not phase trajectories, the necessity to give an answer to the question on the existence of trajectories of such a type is obvious [39]. Moreover, closed trajectories are always key trajectories (at least for systems on two-dimensional manifolds), since the global position of many other phase trajectories depends on their position. The latter fact is explained by the fact that the phase curves composed of the trajectories of dynamical systems on two-dimensional manifolds and contractible into a point along them separate the phase manifold into two parts [39]. If the desired curves exist, then the construction and study of the Poincar´e first return function is the main method for their study. 4.2.1. Problems of existence of monotone limit cycles. Let us consider a closed trajectory of a given vector field in a many-dimensional space and a transversal small area of codimension 1 at a certain point. The first return mapping transforms the vectors of the field applied to the small area into the vectors of the field also applied to this small area. Therefore, there exists at least one eigenvector corresponding to the eigenvalue equal to 1 for this first return mapping. The latter eigenvector is a vector of the field applied to the point of a closed trajectory (cycle). Assume that the vector field is given in the three-dimensional space. Then the transversal small area is two-dimensional, and we can consider the normal vector at the point of the cycle. We can span the osculating plane on the vector of the system field and the constructed normal at the point. In this plane, we can define the coordinates of the vector of the field and the normal vector. Let the transversal small area pass through the normal vector and the binormal vector at the point. Definition. If for each point of the cycle being studied, the trajectories lying near it and intersecting the constructed perpendicular small area tend to the cycle strictly monotonically, then the cycle is said to be monotone. In other words, the strict monotonicity means that the vectors of the field lying near are not parallel to the vectors of the field along the cycle. We can give a number of equivalent definitions of monotone cycles. We restrict ourselves to the following one. Definition. A cycle is said to be monotone if no one of the vectors of the field lying near the vectors of the field along the cycle does not project into the zero vector on the perpendicular small area near any point of the cycle. In particular, we can give the following definition for dynamical systems on the plane. Definition. A limit cycle on the plane is said to be monotone if under its development on a line, the close trajectories tend to the cycle strictly monotonically. Let us consider systems of the form (72) in the strip Π. The following lemma is easily deduced from the definition of the monotone cycle. Lemma 4.4. If F(α) = 0 for α = 0, then there are no monotone limit cycles. Corollary 4.3. If condition (15) holds, then Lemma 4.4 holds on the whole plane R2 {α, Ω}; precisely, there are no monotone limit cycles on the whole plane.
789
Remark. In what follows, we show that for A1 A2 = 0, systems of the form (72) have no closed curves of trajectories on the whole plane (at least for F ∈ Φ). Therefore, the local reasons for the nonexistence of monotone cycles allow us to study the problem of nonexistence of any closed curves of trajectories in what follows. Theorem 4.3. For a vector field on the plane (denoted by v), in a domain D, let there exist equilibrium points such that ind∂D v = 1. If there exists a line L intersecting ∂D and such that the nonzero vector field component v|L on it makes a constant angle with it, then there is no monotone limit cycle of the field such that inside it, there exists an equilibrium point of the field lying on the line L. Theorem 4.3 has a simple and clear meaning and easily extends to the case of higher dimensions. 4.2.2. On absence of closed curves of trajectories contractible to a point on two-dimensional surfaces. In the previous section, we have already partially considered the problems of existence of limit cycles for dynamical systems in connection with the definition of monotone limit cycles. In what follows, we consider the problems of existence of general closed curves of trajectories of dynamical systems contractible to a point along the two-dimensional phase surface. Using the Stokes theorem, we can prove the following assertion (cf. [93, 405]). Theorem 4.4. Let us consider an orientable Riemannian manifold N 2 ⊂ R3 with a vector field v defined on it. Let w be a nonzero field such that (w, v)|N 2 = 0. If there exists a sufficiently smooth function F on the manifold such that (rot F w, n)|N 2 does not alternate the sign in a given simply-connected domain whose boundary is contractible to a point along the manifold N 2 , then in this domain, there is no closed curve of trajectories of the field v contractible to a point along the manifold [259] (here, n is the exterior normal ). In particular, for the absence of closed curves of trajectories on the plane, it is sufficient that there exists a smooth function F such that the divergence of the field F v does not alternate the sign. In each concrete case, this theorem can be strengthened. As a consequence, we present the following lemma. Lemma 4.5. Let us consider system (72) in the strip Π. If π π A1 = 0, F ∈ C 1 − , 2 2 and the quantity d F(α) dα cos α π π is sign-constant for α ∈ − 2 , 2 , then in the strip Π, there is no closed curve of trajectories. Remark. We can prove that under weaker conditions on the function F, there are no closed curves of trajectories for the field of system (72). Moreover, the sign constancy condition of the above derivative is extra. This is proved when the comparison systems for (72) are considered. Remark. By the Liouville theorem on the constant density phase volume preservation, the conditions of Lemma 4.5 are sufficient for the absence of integral invariant preservation. Using the same method, we can prove the absence of closed curves of trajectories for vector fields in Rn integrating the normal fields along 1-paths [131]. 790
4.3. On the Absence of Closed Curves of Trajectories Not Contractible to a Point along the Phase Cylinder. 4.3.1. General assertions on absence of closed trajectories enveloping cylinder for systems having central symmetry. In the previous section, we have considered the closed trajectories of systems of differential equations contractible to a point along the two-dimensional phase surface. Such trajectories compose a part of the trivial component of the fundamental group. This component always exists for two-dimensional smooth manifolds. In contrast to the trajectories considered in the previous section, the closed curves that are not contracted to a point along the phase manifold may not exist. The latter circumstance is related to the fact that the phase manifold topology can give an obstruction to the existence of a nontrivial component for the fundamental group of the manifold. For simplicity, let us consider the case of a two-dimensional cylinder whose topology admits the existence of closed curves not contractible to a point along the cylinder. A circle enveloping the cylinder can serve as an example of such a curve. Moreover, in the next sections, we often consider the dynamical systems on two-dimensional cylinder. In the previous sections, we have already used the term “a curve consisting of phase trajectories.” In this connection, we give the following traditional definition. Definition. A phase characteristic is a curve tangent to the vector field considered. In particular, the curve consisting of the phase curves of the field is a phase characteristic. Definition. The center of symmetry of a vector field given in coordinates (x, y) is a point (x0 , y0 ) such that after the change of variables x0 + x x0 − x ⇒ y0 − y y0 + y, the vector field alternates its sign. The following theorem holds [39, 405]. Theorem 4.5. Let us consider the following autonomous system: ψ = Ψ(ψ, y),
y = Y (ψ, y)
(93)
on the cylinder G = S1 {ψ mod 2π} × R1 {y}, where ψ, Y ∈ C 1 (G). Let there exist a sufficiently smooth function M : G → R such that the quantity div M v(ψ, y) is sign-definite (here, v(ψ, y) = (Ψ(ψ, y), Y (ψ, y))) in a domain Ω enveloping the cylinder. Then Ω contains at most one closed curve of trajectories of system (93) that is not contracted to a point along the cylinder. Moreover, if there exists the center of symmetry x0 for the vector field of system (93) such that no nontrivial characteristic continues through x0 , then system (93) has not even one closed curve of trajectories of system (93). To explain the latter theorem, we note that if there exists the center of symmetry of the field having the above property, then there exist at least two curves of trajectories (if they exist at all). 4.3.2. Remarks from dynamics of a rigid body interacting with a medium. Although the assertions of this section are sufficiently simple, the ideology of searching for phase characteristics enveloping the phase cylinder (see the previous section) is well demonstrated by the following corollary of Theorem 4.5. Corollary 4.4. Let us consider systems of the form (92) on a cylinder. If d F(α) , |h| ≥ A1 max dα cos α then systems of the form (92) have at most one curve of trajectories that envelopes the cylinder. Moreover, if F ∈ Φ, then there are no such curves at all. 791
Although this corollary is obvious, under the condition that F ∈ Φ, systems of the form (92) have no closed characteristics enveloping the phase cylinder for any h = 0 (see Sec. 10). There are infinitely many such curves for h = 0. Remark. When the comparison systems [226,227,298] are examined, we show that for the system describing the body free drag in a medium (this system is given on a cylinder), there is no phase characteristic enveloping the phase cylinder under certain natural conditions. 4.4. On Existence of Poincar´ e Topographic Systems in Dynamics of a Rigid Body Interacting with a Resisting Medium. For studying closed trajectories of dynamical systems arising in the dynamics of a rigid body interacting with a medium, let us apply (and also modify in some cases) the Poincar´e topographical system (PTS) theory [226, 227, 298]. 4.4.1. Poincar´e topographical systems. Poincar´e proposed the (although not quite general) method for searching for closed orbits of the differential equation (91). For this purpose, he was forced to introduce the concept of “topographical systems.” For the first time, he considered an algebraic function F (x1 , x2 ) such that (1) it is sufficiently smooth and bounded in any bounded domain and tends to infinity as one of its variables tends to infinity; (2) it is equal to zero for x1 = x2 = 0 and is positive at other points; (3) it has first derivatives vanishing only for x1 = x2 = 0; (4) for x1 = x2 = 0, the following inequality holds for it:
∂X1 ∂ 2 F − ∂x2 ∂x21
∂X1 ∂X2 − ∂x1 ∂x2
2 ∂2F ∂X2 ∂ 2 F − ∂x1 ∂x2 ∂x1 ∂x22 2 2 ∂ F ∂X1 ∂X2 ∂X1 ∂X2 ∂2F ∂2F < 0. +4 − − ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 ∂x21 ∂x22
This inequality means that the contact curve has an isolated singular point at the origin (for the contact curves, see below and [226, 227, 298]); (5) it is such that the curve defined by the equation X1
∂F ∂F + X2 =0 ∂x1 ∂x2
does not go to infinity. Under Conditions (1)–(5), the equation F (x1 , x2 ) = k yields the so-called topographic system of curves embedded in each other with vertex at the origin. It should be noted that we do not need all of Conditions (1)–(5) in what follows. We take account of only the geometry of contact curve location, trajectories of the dynamical system considered, and curves of the Poincar´e topographical system. In what follows, we consider the systems of the form (72); (92); (86), (87); (79), (80) on the cylinder S1 {α mod 2π} × R1 {ω} (or on the plane S1 {α mod 2π} × R1 {Ω}), or on the plane R2 {α, ω} (or R2 {α, Ω}). Unless otherwise specified, we can assume that conditions (15) and (13) hold. By a PTS we mean a system of closed curves embedded in each other and obtained by using the level surfaces of a nonnegative function such that it vanishes only at the point to which the obtained closed embedded curves converge. Using such a system, we can successfully “catch” closed trajectories of the dynamical system considered: calculating the angle made by the vectors of the field forming the PTS and the vectors of the field of the dynamical system being studied, we can obtain information on the location 792
of trajectories of the vector field being studied. To prove the absence of the closed phase characteristics, it suffice to verify the inequality ∂F ∂F X1 + X2 ≤ (≥) 0. ∂x1 ∂x2 Here F (x1 , x2 ) = const is a family of closed curves and X = {X1 , X2 } is the initial vector field. Therefore, using the latter inequality, we can study the qualitative behavior of trajectories of initial system. Note that the vector field must necessarily be given in certain coordinates. We have already mentioned above that it is possible to define a PTS more correctly, but we do not need this in what follows. Moreover, we are interested only in the geometric properties of the mutual location of PTS curves and phase curves of the field being studied. 4.4.2. Characteristic functions and contact curves of vector fields and dynamics of a rigid body interacting with a medium. The concept of characteristic function of two vector fields on the plane is closely related to the concept of Poincar´e topographical system. The latter function defines a skew-symmetric form on the plane. If F (x1 , x2 ) = const is a family of closed curves, then a system of the explicit Hamiltonian form ∂F ∂F x1 = − , x2 = ∂x2 ∂x1 defines the vector field tangent to the family of PTS curves. Then the latter inequality is equivalent to the inequality X1 Y2 − X2 Y1 ≤ (≥) 0 ∂F ∂F , Y2 = ∂x is the vector field of the system. It is tangent to the PTS curves. in which Y1 = − ∂x 2 1 Let us consider two systems of equations on the plane. These equations are given by fields X = {X1 , X2 } and Y = {Y1 , Y2 } in some coordinates. It is natural to consider the functions χ = X1 Y2 − X2 Y1 , which controls the sign of the sine of the angle between the fields X and Y . Obviously, χ = 0 at the points at which the fields X and Y are tangent and only at these points. The function χ satisfies the following properties:
χ(X, Y ) = −χ(Y, X),
χ(λX, Y ) = λχ(X, Y )
for any real-valued function λ. Definition. The function χ is called the characteristic function of two vector fields, and the equation χ = 0 is called the contact curve for the fields X and Y . As an application of the PTS method to dynamical systems arising in the dynamics of a rigid body interacting with a resisting medium, let us consider systems of the form (72) in the strip Π. Previously (Sec. 4.2), we have shown that under certain conditions, such a system has no monotone limit cycles. It turns out that under the same conditions, this system has no closed curves of phase trajectories. Lemma 4.6. Let A1 A2 = 0. If F(α) = 0 for α = 0 in the strip Π, then a system of the form (72) has no closed curve of trajectories in the strip Π whenever F (0) > 0 and F(0) = 0. Proof. For A1 = 0, system (72) is called system (72) . For system (72) , the origin is a singular point whose topological type is the center. Therefore, there exists a family of periodic trajectories that are arbitrarily close to the points π π ,0 . − ,0 , 2 2 The characteristic function corresponding to systems (72) and (72) is equal to A1 A2
F 2 (α) . cos α 793
The line {(α, Ω) ∈ R2 : α = 0} is a unique contact curve. The characteristic function is positive at other points of the strip Π. Let us proceed by contradiction. Let there exist a closed curve γ0 of trajectories that bounds the domain S0 . Let (α0 , 0) ∈ γ0 , where π 0 < α0 < . 2 Consider the point (α0 + ε, 0), where ε is sufficiently small. A PTS trajectory γε bounding the domain Sε passes through this point. Obviously, emanating from the point (α0 , 0), the trajectory γ0 never attains the domain Sε since the characteristic function is positive almost everywhere. But (α0 , 0) ∈ Sε . We obtain a contradiction. The lemma is proved. Corollary 4.5. If F ∈ Φ, then in the strip Π, there is no closed characteristic of system (72). Let us consider the system (79), (80) in the strip Π. Lemma 4.7. In the strip Π, for σ = 0, the system (79), (80) has no closed curve of trajectories whenever (α) = 0 for α = 0 and also when F (0) > 0 and F(0) = 0. Corollary 4.6. If F ∈ Φ, then in the strip Π, there is no closed characteristic of the system (79), (80). Now let us consider the system (86), (87) in the strip Π. Lemma 4.8. In the strip Π, for σ = 0, a system of the form (86), (87) has no closed curve of trajectories whenever F(α) = 0 for α = 0 and also when s(α)|Π ≥ 0, F (0) > 0, and F(0) = 0. Corollary 4.7. If F ∈ Φ and s ∈ Σ, then in the strip Π, there is no closed characteristic of the system (86), (87). Remark. As was already shown, in the strip Π , a system of the form (86), (87) can have limit cycles under certain conditions. Let us consider the system (86), (87) in the domain O = {Π(0,π) ∩ {(α, ω) ∈ R2 : ω > 0}}. If s( π2 ) = F( π2 ) = 0, then there exists the singular point π 1 , . 2 σ Here Π(0,π) = {(α, ω) ∈ R2 : 0 < α < π}. Lemma 4.9. Let us consider a system of the form (86), in the domain O . For simplicity, let F ∈ Φ, π 1 (87) and let s ∈ Σ. Then around the equilibrium point 2 , σ , in the domain O , there is no closed curve of trajectories whenever the equation ω 2 s(α) cos α −
1 σ ωF(α)s(α) + F(α)s(α) sin α = 0 I I
(94)
holds in the domain O only for α = π2 . Note that relation (94) arises in studying the so-called nontrivial equilibrium states (NES) of the system (86), (87). For NES, see Sec. 6. In general, in studying the system (86), (87), depending on the domain, one of the systems of the form (79), (80) or (79), (80) is used. System (79), (80) is the system (79), (80) for σ = 0. 794
4.5. Contact Curves and Comparison Systems. Remarks on Limit Cycles and Problem of Distinguishing Center and Focus. 4.5.1. Comparison system and studying the topological structure of trajectory location. The PTS method, which was discussed in Sec. 4.4, is a particular case of the method of study using comparison systems. Let us consider two systems of equations on the plane and the characteristic function of vector fields defining them, which, as was mentioned above, control the sign of the sinus of the angle between the vector fields of these systems. Knowing the phase topology of partition into trajectories of one of them, we can analyze the structure of the phase plane of the other system. In particular, the PTS method allows us, e.g., to study the problem of existence of limit cycles [262, 298]. Therefore, we mainly focus on the calculation of the angle between the fields of the systems considered at the same domain of the phase surface. We can calculate the characteristic function for any two vector fields on the plane. In this connection, the system whose qualitative trajectory location is completely known is called the comparison system for a given system. Previously, we have performed the comparison of the vector fields of the systems (72) ; (86), (87); (79), (80). Proposition 4.1. Let us consider systems of the form (79), (80) and (86), (87) on the plane. System (79), (80) is a comparison system for the system (86), (87) as follows: almost everywhere, the angle between the vector of one field and the vector of the other field lies in the limits from 0 up to π, and this angle is equal to zero only on a set of zero measure if all conditions of Lemma 4.9 hold. Indeed, if F ∈ Φ and s ∈ Σ, then by Lemma 4.9, the characteristic function is sign-definite on the whole phase plane. Remark. If conditions (16) and (14) hold, then the angle between the vectors of the fields of the systems (86), (87) and (79), (80) monotonically varies with respect to the parameter B . µ1 = 2mn0 In particular, for µ1 = 0, the latter characteristic function vanishes identically. Remark. Since the system (79), (80) has three topologically different types of phase portrait [260] on the plane, it follows that to study the system (86), (87) in each of the domains of its parameters, we can use its own topological type (see Sec. 6). As a comparison system for the system (86), (87), we can use the system (79), (80) , which describes a conservative system, the physical pendulum on the plane. Remark. The system (86), (87) has no closed curves of trajectories that envelope the phase cylinder whenever all conditions of Proposition 4.1 hold. Proposition 4.2. Let us consider the systems (86), (87) and (79), (80) on the whole plane. The system (79), (80) is a comparison system for the system (86), (87) as follows: almost everywhere in the strip Π, the angle between the vector of one field and the vector of the other field lies in the limits from 0 up to π, and only on a set of zero measure, this vector is equal to zero; in the strip Π , under the condition that π σ F(α) s(α) ≥ for all α ∈ 0, , m cos α I sin α 2 there exists the contact curve of two vector fields considered, which is the topological circle. Proof. Indeed, if F ∈ Φ and s ∈ Σ, then in the strip Π, the contact curve is the origin. Let us translate the origin into the point (π, 0) and rewrite the equation of the contact curve in the form
1 s(α) s(α) F(α) 2 4 sin α + ω 2 cos α. (95) + ω = F(α) σ cos α I I m m 795
Then Eq. (95) of the contact curve, which is said to be nontrivial, can be resolved with respect to ω 2 :
s(α) σ F(α) s(α) F(α) s2 (α) 2 2σω = ± sin α − . (96) + 4σ m m2 I m cos α I sin α In the strip Π, Eq. (96) taken with the plus sign defines only the point (0, 0), whereas it taken with the minus sign defines the point (0, 0) and the contact curve. The nontrivial contact curve is symmetric with respect to both coordinate axes (after its translation from the strip Π into the strip Π) and intersects both axes at a right angle only two times (by the implicit function theorem). This thus proves the proposition. 4.5.2. Some general assertions on PTS and comparison systems. In order to consider the problems of existence of key trajectories (closed phase characteristics, etc.), let us prove a certain general theorem. For this purpose, we note that up to the lines Λ−1 and Λ0 (and also Λ0 and Λ1 ), there exists a family of closed curves that is PTS (integral curves of the system (79), (80) ). Here π Λk = (α, ω) ∈ R2 : α = + πk . 2 Also, let us pose the problem on the existence of closed curves of trajectories for the system (86), (87) in the strip Π . For this purpose, let us prove the following assertion, which generalizes the arguments used in proving Lemmas 4.6, 4.7, 4.8, and 4.9. Theorem 4.6. In a simply-connected domain D of the plane containing an equilibrium point x0 of a sufficiently smooth vector field v1 , let there exist a curve γ x0 connecting two points A, B ∈ ∂D (the points A and B can be infinitely distant) and such that there exists a PTS centered at x0 defined by a sufficiently smooth function V that is continued along γ up to A B, fills the domain K ⊆ D, and has the property (v1 , v2 )|R2 > 0 (where v2 = grad V ) almost everywhere in K possibly except for some curves that do not surround x0 . (Here V = const is a family of PTS curves.) Then in the whole domain D, around the point x0 , there are no closed curves of trajectories of the field v1 . Proof. Let us proceed by contradiction. Let there exist such a curve γ0 bounding a domain S0 x0 . Let {N1 , N2 } = γ ∩ γ0 = ∅ (since x0 ∈ γ), and let the point N1 be a nonsingular initial condition for the motion along the curve γ0 . A closed curve γ¯ from PTS passes through the point N1 , and, moreover, ¯ then there exists ε > 0 (which can be as small as needed) such γ¯ ⊂ K. If the curve γ¯ bounds a domain S, that (1) Nε ∈ γ¯ε ∩ γ, where γ¯ε is a PTS curve; (2) the distance between the points Nε and N1 is equal to ε; ¯ (3) γ¯ε bounds a domain S¯ε ⊃ S. The chosen value of ε is such that in a finite time, moving along the trajectory γ0 with the initial condition N1 , the point leaves the domain S¯ε . Since S¯ε ⊂ K and the inequality (v1 , v2 )|R2 > 0 holds almost everywhere in K, possibly except for some curves not surrounding x0 , it follows that the point with the initial condition N1 never returns to the domain S¯ε ⊂ K ⊂ D. Since S¯ ⊂ S¯ε , we obtain a contradiction to the closedness of the curve γ0 . As was already noted, in the strip Π , for systems of the form (86), (87), under certain conditions, there exists a PTS centered at the point (π, 0) that is continued up to the points ( π2 , 0) and (3 π2 , 0). Moreover, the nontrivial contact curve of the comparison system (PTS) and the fields of the system (86), (87) bound a domain entirely containing the PTS. Therefore, Theorem 4.6 implies the fulfillment of the following lemma. 796
Lemma 4.10. Let us consider the system (86), (87) in the strip Π in the case where the nontrivial contact curve of the comparison system (79), (80) and (86), (87) bound a domain that entirely contains the system the PTS and is continued up to the points π2 , 0 and 3 π2 , 0 . Then in the strip Π , there is no closed curve of trajectories of the system (86), (87). Remark. Let us consider systems of the form (86), (87) under conditions (15) and (13). We have posed above the problem on the existence of closed curves of trajectories in the strip Π , i.e., the curves surrounding the point (π, 0). Also, let us pose a more general problem on the existence of any closed curves of trajectories for the system (86), (87) that are contracted to a point along the phase cylinder. Obviously, the sum of indices of singular points lying inside such curves must be equal to 1. Therefore, such aroundthe points (πk, 0), k ∈ Z. Inside such curves, there cannot be two curves can arise only saddles − π2 + 2πk, 0 and π2 + 2πk, 0 and the point (2πk, 0) simultaneously since the sum of indices is equal to −1 in this case. Inside such curves, there cannot be the points (πk, 0) and (π(k + 1), 0) simultaneously since this is impossible by the central symmetry of the field of the system (86), (87) with respect to the points (πk, 0) and the uniqueness theorem. Only one possibility of existence of such a curve containing more than one singular point inside itself remains. For definiteness, let k = 0. Such a curve can contain the points π π π 1 π 1 ,0 , − ,− , , (0, 0), − , 0 , 2 2 2 σ 2 σ whose sum of indices is equal to 1. But, by Lemma 4.9, this is impossible at least for the reason that a trajectory whose ω-limit set is the point ( π2 , σ1 ) has an infinitely distant point as the α-limit set (see the next section). Therefore, the problem on the existence of closed curves of trajectories for the system (86), (87) that is contracted to a point along the phase cylinder under the fulfilment of Lemma 4.9 reduces to finding such curves in the strip Π around the point (π, 0). As is shown in Sec. 4.1, such curves exist under certain conditions. 4.5.3. Problem of distinguishing the center and the focus and comparison systems. The first distinguishing problem in qualitative theory of ordinary differential equations, the focus and center problem, arises at the point (π, 0) in the strip Π for a dynamical system of the form (86), (87) under conditions (15) and (13) and also for µ1 = µ2 and 0 < µ2 < 2 (see Sec. 4.1). In Sec. 4.1, we show that for In = 0, its solution is a weak focus. Therefore, in a sufficiently small neighborhood of the point (π, 0), all trajectories-spirals approach the point (π, 0) either as t → +∞ or as t → −∞ (here, t is an independent parameter along trajectories). Let us pose the problem of enlarging this neighborhood of the point (π, 0) in the strip Π (on the plane R2 {α, ω}). For this purpose, let us translate the origin into the point (π, 0). After such a change of variables, the system (86), (87) (now being considered in the strip Π) takes the form σ s(α) F(α) cos α − σω 2 sin α + sin α, I m (97) ω 1 σ ω = − F(α) − σω 3 cos α + ωF(α) sin α + s(α) cos α. I I m For simplicity, let us consider system (97) under conditions (16) and (14), which means that F(α) = F0 (α) and s(α) = s0 (α). In the strip Π, under the change according to the formula τ = sin α, |τ | < 1, to system (97), we put in correspondence the equation σn2 √ −n20 τ − σω 3 + σn20 ωτ 2 + 2 0 ω 1 − τ 2 dω √ = . (98) dτ ω − σn20 τ (1 − τ 2 ) − σω 2 τ + σn2 0 τ 1 − τ 2 α = ω −
797
In Eq. (98), we have already taken the condition µ1 = µ2 into account, which is equivalent to the condition B σn0 . = mn0 2 Formally, let τ = x and ω = y. System (97) whose phase trajectories coincide with integral trajectories of Eq. (98) and which is considered in the strip {(x, y) ∈ R2 : − 1 < x < 1} σn0 x 1 − x2 , 2 σn20 y = −n20 x − σy3 + σn20 yx2 + y 1 − x2 . 2 Let us consider the system
reduces to the form
x = y − σn20 x + σn20 x3 − σy2 x +
(99)
3σn20 3 σn20 x+ x − σy2 x, 2 4 (100) σn20 3σn20 2 2 3 y − σy + yx , y = −n0 x + 2 4 which is a comparison system for system (99). As was already noted, the characteristic function of the pair of systems (99) and (100) is a skew-symmetric bilinear function. For the ordered pair of systems X and Y, we denote by χ(X, Y) its characteristic function. x = y −
Proposition 4.3. For σn0 < 2, in the strip {(x, y) ∈ R2 : − 1 < x < 1}, the characteristic function χ((99), (100)) is positive-definite (it is equal to zero only at the origin). Indeed,
x2 σn20 2 2 2 2 2 [y − σn0 xy + n0 x ] 1 − − 1−x . χ((99), (100)) = 2 2 Note that the right-hand sides of system (100) differ from those of system (99) only by terms of the fifth order of smallness in ρ = (x2 + y2 )1/2 , i.e., by terms of order O(ρ5 ). In connection with the remark on the representation of the vector field of system (99) near the point (0, 0) just made, the following proposition holds. Proposition 4.4. The point (0, 0) is a complicated stable focus of system (100) for σn0 < 2. Let us pose the following question: what is the relation between the sign of the characteristic function considered as a form of an ordered pair of systems on the plane and the angle of turn from the field of one system to another? For this purpose, let us consider the system x = y − σn20 x + σn20 x3 − σy2 x + λx 1 − x2 , (101) y = −n20 x − σy3 + σn20 yx2 + λy 1 − x2 , which depends on the parameter λ > 0. The vector field of system (101) has some strict monotonicity property (see below) under certain conditions and for σn0 < 2 in any domain without singular points. If, for λ = λ1 , we denote system (101) by (101) and by (101) for λ = λ2 , then the following relation holds: χ((101) , (101) ) = (λ2 − λ1 )[y2 − σn20 xy + n20 x2 ] 1 − x2 . Therefore, for λ2 > λ1 , the latter function is positive-definite in the strip {(x, y) ∈ R2 : − 1 < x < 1} (it is equal to zero only at the origin). But, as is easily understood, for system (101) , the origin is a “more unstable” singular point than the origin for system (101) . Therefore, for λ2 > λ1 , the vector field of system (101) turns around the vector field of system (101) by a positive angle. 798
Before speaking about the character of trajectories of system (99), let us formulate an auxiliary assertion, which is of independent interest. To find a suitable comparison system in order to study the existence of limit cycles, the problems of distinguishing the center and the focus, etc., it is not necessary to have a PTS centered at a given singular point. The desired comparison system can have either an attractive or repelling singular point. In a domain D containing a unique singular point of system (A) which is given in the plane for simplicity, let one pose the problem of distinguishing the center and the focus. In the same domain, let system (B) have the same unique singular point x0 . For definiteness, let us consider the so-called negatively orientable systems in which trajectories go around the point x0 counterclockwise.The positively orientable systems can be considered analogously. Lemma 4.11. Let the domain D be attracted (repelled ) by the point x0 of the negatively orientable system (B). If the characteristic function χ((A), (B)) is positive- (negative-)definite in the domain D, then the domain D is attracted (repelled) by the point x0 of (negatively orientable) system (A). Lemma 4.11 is of obvious geometric character. Indeed, the vector field of system (B) is turned with respect to the vector field of system (A) by a nonnegative (nonpositive) angle. An analogous assertion can also be formulated for stable and unstable limit cycles, etc. Therefore, as system (A), we take system (99) (which is negatively orientable), and as system (B), we take system (100) (which is also negatively orientable). There arises the question about the size of the domain D from Lemma 4.11. Proposition 4.5. For σn0 < 2, in the strip {(x, y) ∈ R2 : − 1 < x < 1}, system (100) has a first integral, which is simultaneously a transcendental function in the strip and a meromorphic function in the set {(x, y) ∈ R2 : − 1 < x < 1} without origin. The latter point is a unique essentially singular point of this first integral. Proof. After the change of variables √
√ 3 3 n0 x − y = u, n0 x + y = v 2 2 system (100) reduces to the equation
7 7 v σ σn0 σ u σn0 2 2 √ + − −√ +v − √ + +√ uv + dv u v = 0. du u 12 12 2 3 12 3n0 2 3 12 3n0 After the change u = tv, v2 = p, v = 0, the equation takes the form
√ σ 2 dp[C1 t + C2 ] + 2p C1 t − 1 − u 3 tp = 0. n0 Here we have introduced the notation
√ C1 = 7 + 2 3σn0 ,
√ C2 = 7 − 2 3σn0 .
Using the change p = q −1 , we reduce the latter equation to a linear inhomogeneous equation of the form √ σ dq [C1 t2 + C2 ] + 2[1 − C1 t]q + 8 3 t = 0 (102) dt n0 The general solution of the linear homogeneous equation has the form q0 (t) = C
C1 t2 + C2 . exp C21 arctan C2 t/C1
Varying the constant C, we have a differential equation that allows us to obtain a meromorphic solution of Eq. (102). Singularities arise only for u = v = 0, i.e., when the value t is not found. But the latter equation also defines the origin in the plane R2 {x, y}. 799
Corollary 4.8. For system (100), the attraction domain is the whole strip {(x, y) ∈ R2 : − 1 < x < 1}. Corollary 4.9. In the strip {(x, y) ∈ R2 : − 1 < x < 1}, system (99) has no closed characteristics. B < σn2 0 , in the strip Π , systems of the form (86), (87) Corollary 4.10. For σn0 < 2 and also for mn 0 have no simple and complicated limit cycles and also any closed curves composed of trajectories.
4.6. On Trajectories Having Infinitely Distant Points of the Plane as Limit Sets. 4.6.1. Examples from dynamics of a rigid body interacting with a medium. In this section, we consider the problems of existence and uniqueness of trajectories of dynamical systems (91) on the plane that have infinitely distant points as their α- and ω-limit sets [264]. Therefore, on the Riemann or Poincar´e spheres, the north pole is the limit sets of these trajectories. Even by definition, such trajectories are key trajectories since an infinite distant point is always a singular point. To begin with, we consider systems of the form (79), (80) and (86), (87). Lemma 4.12. Let us consider the system (79), (80) on the set Π ∩ {(α, ω) ∈ R2 : ω > 0}. Then for any sufficiently smooth function F, there exists a unique trajectory going to infinity (having the point (−0, +∞) as the ω-limit set). Proof. Let us complement the phase plane R2 {α, ω} by the infinitely distant point. Then we obtain the extended phase plane R2 {α, ω}. Let us map the domain Π ∩ {(α, ω) ∈ R2 : ω > 0} into the Riemann or Poincar´e sphere. In a neighborhood of the north pole of the sphere, there exist new coordinates (α, y), y = ω1 into which the old coordinates of the extended phase plane transform by a nonsingular transformation. In the new coordinates (α, y), the system (79), (80) is equivalent to the equation dα = dy
y + σI y2 F(α) cos α + σ sin α
y4 I F(α)
− σy cos α + σI y3 F(α) sin α
.
In this case, the trajectories of this equation are parameterized in another way than the trajectories of the system (79), (80). It is seen that this equation has the singular point (0, 0) corresponding to the infinitely distant point (0, +∞) of the system (79), (80). It is easy to verify that the point (0, 0) is a hyperbolic saddle; this proves the lemma. The following lemma is proved analogously. Lemma 4.13. Let us consider system (86), (87) on the set Π ∩ {(α, ω) ∈ R2 : ω > 0} Then for any sufficiently smooth functions F and s, there exists a unique trajectory going to infinity (having the point (0, +∞) as its ω-limit set). 4.6.2. Existence and uniqueness of trajectories going to infinity. Let us consider an arbitrary autonomous system of differential equations of the form (91) on the plane. To this system, we put in correspondence an equation whose phase trajectories are parametrized in another way, and the latter are also mapped from the extended phase plane into the Riemann (or Poincar´e) sphere. As was already noted, in this case, the infinitely distant points pass to the north pole of the spere. 800
Theorem 4.7. (1) If after the change of the phase variables (x1 , x2 ) → (x1 , y), where y = x12 , there arises a singular point (x01 , 0) for the equation defined on the sphere, then the system considered has a trajectory tending to the line {(x1 , x2 ) ∈ R2 : x1 = x01 } and having the infinitely distant point as the limit set. (2) If after the change of phase variables (x1 , x2 ) → (y, x2 ), where y = x11 , there arises a singular point (0, x02 ) for the equation defined on the sphere, then the system considered has a trajectory tending to the line {(x1 , x2 ) ∈ R2 : x2 = x02 } and having the infinitely distant point as the limit set. Remark. The number of trajectories going to infinity is found through the topological type of the infinitely distant singular point. In particular, in systems (79), (80); (86), (87), there exists a unique trajectory going to infinity since the infinitely distant singular point is a saddle (of course, if we map not the plane but the phase cylinder). Remark. There can exist phase trajectories going to infinity on the phase plane along which both phase variables infinitely increase. In this case, after the change x1 = y11 , x2 = y12 , when studying the topological type of the north pole of the sphere, which is always a singular point, one can try to prove the existence and uniqueness of trajectories approaching lines of the form A1 x1 + A2 x2 + A3 = 0, where A1 A2 = 0. Indeed, in this case, the trajectory always tends to the north pole of the sphere at a certain angle, which corresponds to the trajectory on the plane tending to the line that has a nonzero finite inclination coefficient. 4.6.3. Elements of monotone vector field theory. Let us consider a family of sufficiently smooth vector fields vε in a domain D of a two-dimensional oriented Riemann surface. In the tangent space Tq D of each point q ∈ D, we can measure the angles between the vectors of the family considered [33]. Definition. A one-parameter family of fields vε (ε ∈ E) has the monotonicity property in D if, for any points q ∈ D, ε1 ∈ E, and ε2 ∈ E, in the tangent space Tq D, the angle between the vectors vε1 , vε2 ∈ Tq D is a monotone function of the difference ε2 − ε1 of parameters, and, moreover, the orientation of variation of the angle is preserved. If the monotone dependence considered is strict, then we say that vε has the strict monotonicity property (cf. [67–70]). Theorem 4.8. Let a field vε have the monotone property in a domain D of the plane R2 . Let x0 be a nonsingular initial condition for the phase trajectory of the field vε for all ε ∈ E. Then for any ε ∈ E, the limit set of the trajectories emanating from x0 is the set γ0 ; moreover, if {A, B} = ∂γ0 , A is the limit set of the trajectory of the field vε1 , and B is the limit set of the trajectory of the field vε2 , ε1 < ε2 , then ε ∈ (ε1 , ε2 ) if and only if there exists a set C that is the limit set of the trajectory of the field vε , and, moreover, when ε increases, the limit set monotonically moves from A to B. (Here we simultaneously speak of either the α- or ω-limit sets of the trajectories of the family.) Moreover, the desired phase trajectory is unique if the monotonicity property is strict. A scheme of proof. We can assume that for any ε, the set γ0 consists of ω-limit sets. According to the theorem on continuous dependence of solutions on initial conditions and right-hand sides of equations, under a small variation of the parameter ε, the limit set remains in a close neighborhood of the initial set (if the set γ0 is simply connected). If the latter set is multiconnected, then we sequentially consider each of the connected components. Since the monotonicity property holds, applying the comparison system theory, we exclude the nonmonotone dependence of the trajectory on the parameter ε. 801
Let the system have the strict monotonicity property. We perform the proof proceeding by contradiction. For a point M ∈ γ0 , let there exist at least two parameters ε1 and ε2 for which the trajectories of the fields vε1 and vε2 tend to the point M . Then the trajectories of all fields vε¯, ε¯ ∈ [ε1 , ε2 ], tend to the point M (because of the monotonicity property). Since the monotonicity property is strict, for any δ > 0, the system with vector field vε+δ (ε + δ ∈ E) is a comparison system for vε . It is easy to understand that the trajectory of the field vε+δ emanating from a nonsingular initial condition never intersects the corresponding trajectory of the field vε emanating from the same initial condition. Therefore, the trajectories of the fields vκ1 and vκ2 have different limit sets, and, moreover, ε1 < κ1 < κ2 < ε2 . We obtain a contradiction. In a similar way, we can prove the qualitatively different lemma, which holds on any smooth two-dimensional oriented manifolds. Lemma 4.14. Let us consider a family of fields vε (ε ∈ E) in a domain of a sphere S2 [264] that has the following form. The south pole (S) and the north pole (N) of the sphere are saddles. Let this family of vector fields have the strict monotonicity property, and for a certain ε1 , the south pole is the ω-limit set of a trajectory emanating from the south pole, whereas, for a certain ε2 > ε1 , the north pole is the ω-limit set of a trajectory emanating from the north pole. Moreover, let both situations considered be homoclinic situations on the sphere when there exists only one equilibrium point (except for N and S) that is contained in the domain bounded by the above separatrices. There are no other nontrivial limit sets in this domain of the sphere. Then there exists a unique value ε = ε0 ∈ (ε1 , ε2 ) of the parameter such that the trajectory emanating from the south (north) pole enters the north (south) pole (this is a heteroclinic situation on the sphere). Proof. Uniqueness. We perform the proof proceeding by contradiction. Let two parameters ε¯ and ε¯ have the above property. Then by the strict monotonicity property, all parameters from the interval (¯ ε, ε¯) have the above property. Arguing as in Theorem 4.8, we arrive at a contradiction to the monotonicity property. Existence. By what was proved above, there exists a unique value ε = ε0 of the parameter such that for ε < ε0 , one homoclinic situation on the sphere is realized, and for ε > ε0 , the other homoclinic situation is realized. Let us perform the proof proceeding by contradiction. For ε = ε0 , let one of the homoclinic situations be realized. Then there exists a neighborhood Uεδ0 = {ε : |ε − ε0 | < δ}, of ε = ε0 such that for any ε ∈ Uεδ0 the same homoclinic situation is realized. We obtain a contradiction. The lemma is completely proved. Remark. We have obtained one more method for proving Lemmas 4.12 and 4.13. Indeed, the desired fields satisfy the conditions of Lemma 4.14 since the infinitely distant point is projected on the north pole of the Riemann (Poincar´e) sphere, whereas the point (− π2 , 0) is projected on the south pole. 4.7. Periodic and Poisson Stable Trajectories in Phase Spaces of Dynamical Systems. In systems of ordinary differential equations, there often exist infinite families of closed trajectories that are everywhere dense in certain sets. Moreover, nonclosed trajectories also turn out to be everywhere dense in a certain domain of the phase space. Here along with the concept “everywhere dense in a set,” there arises a more concrete concept of everywhere dense around itself. The latter property is classically called Poisson stability [28, 29, 48, 226, 227, 273, 282, 314, 316, 368, 370, 371, 390, 430]. Recall that a trajectory in the phase space is Poisson stable if in a finite time, it returns to any sufficiently small neighborhood of any its point. Sufficient conditions for the existence of Poisson stable trajectories are formulated in the following theorem. Theorem 4.9. Let us consider a system of differential equations x˙ = f (t, x, µ) 802
(103)
depending on the parameter µ ∈ M ⊆ R1 in a domain D ⊆ Rn+1 {x, t}. For certain initial conditions (x0 , t0 ) ∈ D, let the projection of the integral trajectory (x∗ (t), t) with these initial conditions and the parameter µ = µ0 on the space Rn {x}be a nonclosed curve that is continued to the whole time axis. If in any neighborhood Uµ0 of the point µ = µ0 ∈ M , there exists µ, such that the projection of the integral trajectory x(t, µ) with arbitrary initial conditions (x1 , t1 ) ∈ (x∗ (t), t) on the space Rn {x} is a closed curve, then the curve x∗ (t), t ∈ R, belonging to the space Rn {x} is everywhere dense around itself (Poisson stable). Proof. Consider an arbitrary point x1 of the projection of a nonclosed integral curve x∗ (t) and the projection of the solution to system (103) with the initial condition (x1 , t1 ) ∈ (x∗ (t), t) for µ = µ0 . Since this solution is continued to the whole time axis, by the theorem on the continuous dependence of solution on right-hand sides and parameter, for any ε > 0, there exist µ1 sufficiently close to µ0 and a positive number T1 such that the following condition holds for t1 ≤ t ≤ T1 : |x∗ (t) − x0 (t)| < ε. Here x∗ (t) is the solution of system (103) for µ = µ0 with the initial condition (x1 , t1 ) and x0 (t) is the solution of system (103) for µ = µ1 with the same initial condition. Therefore, for any ε > 0, in the ε-neighborhood Uxε1 of the point x1 ∈ Rn {x}, there exists a solution of system (74) for µ = µ1 and certain T1 > 0. This implies the density of the trajectory x∗ (t) belonging to the space Rn {x} near itself. Moreover, the space Rn {x} is the projection of the space Rn+1 {t, x}. Theorem 4.9 is proved. Remark. By closed curves in the space Rn {x} we mean the projections of periodic solutions of system (103) considered as integral curves from the space Rn+1 {x, t} on the space Rn {x} [370, 371]. Remark. If we consider the closure Z1 of a Poisson stable trajectory x∗ (t) as a set in the space Rn {x}, then the family of closed trajectories considered is everywhere dense in the set Z1 . In turn, if we consider the closure Z2 of the set of closed trajectories as a set in the space Rn {x}, then the Poisson stable trajectory x∗ (t) is also everywhere dense in the set Z2 . Corollary 4.11. The sets Z1 and Z2 coincide. The direct and inverse inclusions follow from the density of the family of closed curves and the Poisson stability. As is shown in Secs. 6 and 8, some nonautonomous systems in R3 and R4 have Poisson stable trajectories. 4.8. Spatial Poincar´ e Topographical Systems and Comparison Systems. 4.8.1. Topographical systems in the spatial case. The spatial PTS can be defined analogously to the PTS on the plane. In this case, (nondegenerate) level hypersurfaces of codimension 1 in the space Rn compose a topographical system of hypersurfaces embedded in each other and having the vertex at a singular point [226, 227, 287, 288, 298, 304, 305, 333]. Theorem 4.10. In a simply connected domain D of the space Rn containing a unique equilibrium point x0 of a sufficiently smooth vector field v1 , let there exist a hypersurface Γ x0 prolonged up to the boundary ∂D and intersecting it along a surface γ (the surface γ can be infinitely distant) such that there exists a PTS centered at x0 , defined by a smooth function v, prolonged along Γ to γ, filling a domain K ⊆ D, and having the property (v1 , v2 )|Rn > 0 (v2 = grad V ) almost everywhere in K, probably except for some hypersurfaces that do not contain x0 inside themselves. (Here v = const is the family of hypersurfaces of the PTS.) Then the whole domain D contains no closed curve consisting of trajectories of the field v1 and intersecting the hypersurface Γ. 803
Proof. We proceed by contradiction. Let there exist such a curve γ0 , {N1 , N2 } = Γ ∩ γ0 = ∅, and let the ¯ point N1 be a nonsingular initial condition for the motion along the curve γ0 . A closed hypersurface Γ ¯ ¯ from PTS passes through the point N1 , and, moreover Γ ⊂ K. If the hypersurface Γ bounds a domain S, then there exists ε > 0 (which we diminish, if necessary) such that ¯ ε ∩ Γ, where Γ ¯ ε is a hypersurface of the PTS; (1) Nε ∈ Γ (2) the distance between the points Nε and N1 is equal to ε; ¯ ¯ ε bounds a domain S¯ε ⊃ S. (3) Γ The chosen value of ε is such that in a finite time, moving along the trajectory γ0 with the initial condition N1 , the point leaves the domain S¯ε . Since S¯ε ⊂ K and the inequality of the theorem holds almost everywhere in K, probably except for some hypersurfaces that do not contain x0 inside themselves, the point with the initial condition N1 never returns to the domain S¯ε ⊂ K ⊂ D. Since S¯ ⊂ S¯ε , we arrive at a contradiction to the closedness of the curve γ0 . Recall that in a number of cases, spatial PTS’ successfully help us to solve the problem of distinguishing the center and the focus. In the latter case, it is not necessary to have a PTS centered at a given singular point. The comparison system can have either an attracting or repelling singular point. In a domain D containing a unique singular point of system (A), which is given on the plane, let us pose the problem of distinguishing the center and the focus. In the same domain, let system (B) have the same unique singular point x0 . In the spatial case, the characteristic function is constructed as follows. Let v1 and v2 be two smooth vector fields in the space Rn . According to the field v1 , we (nonuniquely) construct a normal smooth vector field n. In each concrete case, the field n is constructed from the reasons that allow us to obtain a signdefinite characteristic function in what follows. The latter is defined as the inner product χ = (n, v2 ). Lemma 4.11 with certain corrections also holds in the spatial case. 4.8.2. Even-dimensional case. In the even-dimensional case, the characteristic function has the most natural form. Example. Let us consider a system of weakly coupled pendulums, i.e., the conservative system with small (nonconservative) additions defined by a field {X1 , X 1 , . . . , Xn , X n } in coordinates x = {x1 , x1 , . . . , xn , xn } of the following form: Xi = −xi + εFi (x),
X i = Gi (x) + εF i (x), dG(0) ≥ 0.
Then it is natural to choose the characteristic function in the form n (Xi Y i − X i Yi ), χ= i=1
where the vector field Y of the comparison system has the form Y = {Y1 , Y 1 , . . . , Yn , Y n },
Yi = −xi , Y i = Gi (x).
Also, in the spatial case, we can present the corresponding assertions similar to two-dimensional PTS’ and comparison systems. We present them in Secs. 8 and 9, where the spatial problems of motion of bodies in a resisting medium and oscillation theory are studied. 5. Relative Structural Stability and Relative Structural Instability of Various Degrees The present section is devoted to the study of problems of relative structural stability (relative roughness) for dynamical systems considered not on the whole space of the dynamical systems but on certain of its subspace. In this case, the deformation space of (dynamical ) systems neither coincides with the whole space of admissible deformations. In particular, we consider system of differential equations arising in the dynamics of a rigid body interacting with a resisting medium. We show their relative roughness and also their relative nonroughness of various degrees under certain conditions. 804
The rough (structurally stable) systems can be considered as the simplest and most numerous dynamical systems in the corresponding space of dynamical systems. Indeed, the rough systems are distinguished by conditions of inequality type, and, therefore, it is natural to consider them as the most general case. We can find a far-reaching analogy between the rough dynamical systems and the functions of a certain variable having only simple roots and also the curves without singularities considered in a finite part of the plane [140]. In particular, this analogy is very fruitful for elaborating effective methods for the qualitative study. In a number of problems, it is of interest to consider the relative roughness, precisely, the roughness with respect to a certain class of dynamical systems, i.e., with respect to a certain subset of the dynamical system space. We can use the concept of relative roughness in distinguishing the simplest nonrough systems, i.e., the systems of the first nonroughness degree and also in classifying nonrough systems with respect to the complexity degree or to the nonroughness degree. Note that from the viewpoint of such a classification of nonrough systems, the conservative systems are systems of infinite nonroughness degree, i.e., the systems of the nonroughness degree higher than any finite nonroughness degree. Therefore, the conservative systems are very “rare” systems from the viewpoint of such a classification. However, considering the class of conservative (or Hamiltonian) systems, we can introduce the concept of system roughness with respect to this class. In fact, such a concept (without term “roughness”) was used by Poincar´e. The systems of the first nonroughness degree can be defined as systems that are relatively rough in the set of (relatively) nonrough systems (the precise definition is given in what follows). A nonrough (relatively nonrough) vector field can be topologically equivalent to a rough (relatively rough) vector field. For example, on the two-dimensional sphere, there can be the situation in which a vector field is (absolutely) nonrough but topologically equivalent to a rough vector field (for the topological equivalence, see below). In the latter case, the main reason for the nonroughness is the derivative nondegeneracy near the limit set [19]. In the case of analytic dynamical systems, requiring the existence of no less than five derivatives of the dynamical system right-hand sides, we can define dynamical systems of the second nonroughness degree as systems relatively rough in the set of nonrough systems that are not systems of the first nonroughness degree. Quite analogously, we can define dynamical systems of the 3rd, 4th, . . . , nth degrees of nonroughness. The definition is introduced inductively. In the case of dynamical systems with analytic right-hand sides, we introduce the concept of closeness of systems. Therefore, in what follows, we say that a dynamical system is a system of the nth nonroughness degree in a closed domain if it is a nonrough system that is not a nonrough system of degree less than or equal to n − 1 and if it is a relatively rough system in the set of nonrough systems that are not nonrough systems of degree less than or equal to n − 1. 5.1. Definition of Relative Structural Stability (Relative Roughness). The classical definition of roughness [10, 15, 17–19] and also the definition given in [194] operate with two objects, precisely, with the classes of dynamical systems and the space of system deformations endowed with its own topology. For the first time, the definition of roughness of a dynamical system on the plane was given under a certain additional assumption on the set of dynamical systems considered. Precisely, it was additionally assumed that the boundary of the domain in which the system is considered is a cycle without contact for trajectories of this system, i.e., it is a simple smooth closed curve having no contacts (not tangent to the system trajectories). Obviously, in this case, the curve is a cycle without contacts for trajectories of any system sufficiently close to the one considered. Although this assumption strongly restricts the class of dynamical systems considered, the meaning of the system roughness concept is preserved, whereas the definition of roughness is considerably simpler than that under general assumptions on the domain boundary. 805
We can introduce the concept of roughness in such a way that it does not prohibit the existence of nonrough trajectories lying on the domain boundary. But this does not correspond to the matter of the nonroughness concept. The introduction of the roughness concept without special assumptions on the domain boundary seems to be natural and necessary from various viewpoints [39]. The concept of topological equivalence of dynamical systems [19, 27] is laid as a base of the concept of roughness (including that of nonroughness of various degrees). Let X r (M ), r ≥ 1, be the space of C r -vector fields on a compact manifold M equipped with the r C -topology. Two vector fields X, Y ∈ X r (M ) are said to be topologically equivalent if there exists a homeomorphism h : M → M that transforms the trajectories of the field X into the trajectories of the field Y and preserves their orientation; the latter condition means that if p ∈ M and δ > 0, then there exists ε > 0 such that if 0 < t < δ, then hXt (p) = Yt h(p) for a certain t ∈ (0, ε). We say that h is a topological equivalence between X and Y . Therefore, we have defined the equivalence relation on Xr (M ). Another and a more strong relation is the conjugacy of flows of vector fields. Two vector fields X and Y are said to be conjugate if there exists a topological equivalence h preserving the parameter t; this means that hXt (p) = Yt h(p) for all p ∈ M and t ∈ R. Remark. The conjugacy of flows is a very strong relation, since, as is known, an expanding cycle (as a function of the parameter) of autonomous systems changes its period as a rule [19,27], and, as is known, closed trajectories pass into closed trajectories under the topological equivalence. Along with the closeness of the system considered with its deformation in a certain topology, the definition given by Andronov and Pontryagin [19] requires closeness to the identity of the homeomorphism through which the topological equivalence of two latter systems is realized, whereas the definition given by Peixoto [216–218] does not require such a closeness. If a system is Andronov–Pontryagin rough, then it is also Peixoto rough. Moreover, the necessary and sufficient conditions for the Andronov–Pontryagin roughness coincide with those for the Peixoto roughness. The latter definition has the following advantage: it directly implies the fact that in the space of dynamical systems, the rough systems fill domains. Under the first definition, it is necessary to prove this fact basing on the necessary and sufficient conditions for the roughness. Let V be a sufficiently small neighborhood of the vector field X considered in X r (C r ). As was already briefly mentioned, in the initial definition of roughness given by Andronov and Pontryagin, it is also required that, for a sufficiently small neighborhood V, the homeomorphism that realizes the topological equivalence between X and Y can be made arbitrarily close to the identity homeomorphism in the C 0 -topology (i.e., it shifts points of M arbitrarily little). Since the variant of this requirement was suggested by Peixoto, where it is necessary to make clearer which variant of roughness is meant, one speaks of the Andronov–Pontryagin roughness and the Peixoto roughness. However, at present, it is not clear whether or not these variants are indeed different from each other and whether or not one of them has an essential advantage as compared with the other. The presented definition depends on r. In the case where it is necessary to show this dependence explicitly, we can speak of the roughness in the class C r [216–218]. Up to now, we have spoken of the global properties of vector fields on manifolds. We can analyze the local topological behavior of trajectories of vector fields [216]. For vector fields from a certain open dense subset in the space X r (M ), we can describe the behavior of trajectories in a neighborhood of each point of the manifold considered. Moreover, the local structure of trajectories does not change under small perturbations of the field (the so-called local roughness). In this way, we obtain a complete classification through the topological conjugacy. Remark. It seems that only in the local case is the topological conjugacy relation constructive since, in the global case, it implies the existence of very strong conditions. 806
In higher dimensions, the set of rough vector fields remains wide but no longer everywhere dense. Here there exist rich and more complicated phenomena that are preserved under small perturbations of the initial field. Even for rough fields, the structure of trajectories of limit sets is far from clear, and its description remains a field of active studies up to now. In parallel to definitions given above, in [15, 222–224], the authors studied low-dimensional rough systems, and in [21,24], the theory of Anosov systems was studied; for the latter, the concept of roughness turned out to be natural. In accordance with the classical definitions of structural stability, in [222–224], the criteria for the latter are discussed for linear nonautonomous systems, as well as for classes of nonlinear systems. The conditions for the structural stability for low-dimensional systems are formulated as the Smale conjectures for large dimensions [432, 433]. In recent time, there also arise several modified definitions of roughness. They all have the following common property: the deformation of dynamical systems considered on a certain manifold M n is taken on the whole space χ(C r ) of smooth vector fields in the C r -topology (most often, r = 1). We consider vector fields (dynamical systems) that are deformed not on the whole class χ(C r ) of fields but on certain its subclass χ(B) defined by a function class B ⊂ C r . Definition. A vector field v on a manifold M n ia said to be relatively structurally stable (relatively rough or rough with respect to the class X(B) of fields defined by the function class B) if for any neighborhood T of the homeomorphism 1M n in the space of all homeomorphisms with C 0 -topology, there exists a neighborhood U ⊂ χ(B) of the vector field v such that the latter is equivalent to any vector field from U ⊂ χ(B) via a certain homeomorphism from T . Note that the closeness of vector fields is understood in the C 1 -topology, whereas the closedness of the homeomorphism is understood in the C 0 -topology. In this case, we speak not of the conjugacy but of the equivalence. Also, note that in the definition presented above, the following is still important: (1) a sufficient smallness of the homeomorphism realizing the equivalence; (2) the C 1 -topology in the space of vector fields considered. 5.2. Relative Structural Instability (Relative Nonroughness) of Various Degrees. Similar to the definition of a vector fields of the first nonroughness degree, we can define fields of the first relative nonroughness degree considering deformations of fields in a subspace χ(B) of the space of all vector fields. Definition. A vector field v on a manifold M n is called a vector field of the first relative nonroughness degree if it is not a relatively rough vector field, if and for any neighborhood T of the homeomorphism 1M n in the space of all homeomorphisms with C 0 -topology, there exists a neighborhood U ⊂ χ(B) of the vector field v such that the field v is topologically equivalent to any field from U ⊂ χ(B) that is not relatively rough via a certain homeomorphism from T . Note that the closeness of vector fields in this case is understood in the C 3 -topology. In a similar way, we can define vector fields that are fields of the nth relative nonroughness degree. In this case, we use the C 2n+1 -topology in the space of vector fields. Definition. A vector field v on a manifold M n is called a vector field of the nth relative nonroughness degree if it is not a relatively nonrough vector field and a relatively nonrough vector field of degree less than or equal to n − 1, and if for any neighborhood T of the homeomorphism 1M n in the space of all homeomorphisms with C 0 -topology, there exists a neighborhood U ⊂ χ(B) of the vector field v such that the field v is topologically equivalent to any field from U ⊂ χ(B) that is not a relatively rough or relatively nonrough vector field of degree less than or equal to n − 1 via a certain homeomorphism from T . 5.3. Examples from Dynamics of a Rigid Body Interacting with a Medium and Oscillation Theory. Let us consider dynamical systems arising in the plane (and further in the spatial) dynamics of 807
a rigid body interacting with a resisting medium. By the cyclicity of some phase variables, the general sixth-order system (for the case of aplane-parallel motion) admits the separation of the independent third-order subsystem (8)–(10). In turn, in the latter system, by the known method, the second-order dynamical system is distinguished; for the latter system, a detailed analysis of various phase portraits that it admits was carried out. 5.3.1. Zero mean variable dissipation systems in rigid body plane dynamics. Such systems have the following common property: as a rule, zero mean variable dissipation systems have additional symmetries and separatrices connecting hyperbolic saddle equilibrium states. Therefore, such systems cannot be (absolutely) structurally stable. Since we consider the deformations of such systems only on a certain subset of all systems defined by a function subclass (right-hand sides) that allows us to preserve all system symmetries, the systems considered remain relatively rough in certain parameter domains. Example. Let us consider systems of the form (72) under condition (15). Lemma 5.1. System (72) is relatively structurally stable. Moreover, any two systems of the form (72) are topologically equivalent. A scheme of proof. Define the space χ(Φ) of vector fields corresponding to system (72); in this case, the function F varies in the whole class Φ. In this case, the system parameter space is infinite-dimensional. Lemma 5.1 follows from the following remarks. (1) For any F ∈ Φ, the phase portrait of system (72) has the same topological type. (2) In each of the domains of the phase cylinder (oscillatory and rotational) (see Fig. 3), we construct its own topological equivalence: on “key” separatrices (see Sec. 6), these equivalences “are concatanated.” (3) For example, in the oscillatory domain, the equivalence is constructed as follows. We construct not only the equivalence, a homeomorphism h of the phase cylinder, but its conjugation. In the oscillatory domain, there exist only two singular points, (0, 0) and (π, 0) (the first of them is repelling, and the second is attracting). Therefore, consider two systems (72) for functions F1 (α) and F2 (α). Denote by g1t and g2t the corresponding phase flows of the phase cylinder. Let us claim that a homeomorphism h transforms the origin into itself. Consider a small circle S1 around the origin. We can choose it to be transversal to both fields of systems (72) for F = F1 and F = F2 simultaneously. Let us define h(p) = p (with accuracy up to a linear contraction or expansion) for all p ∈ S1 so that h(p11 ) = h(p12 ) and h(p21 ) = h(p22 ). Here p1k , p2k , k = 1, 2, are two points on the circle S1 such that separatrices of the field of system (72) for F = Fk emanating from the origin and entering the saddles S−1 and S0 (in the strip Π) pass through them. If q is not the origin, then there exists a unique t ∈ R such that g1t (q) = p ∈ S1 . We set h(q) = g2−t (p) = g2−t g1t (q). It is directly seen that h is continuous and has a continuous inverse. (4) According to the constructed mapping h, the point (π, 0) passes to the point (π, 0) by continuity. Corollary 5.1. Under condition (15), system (72) is topologically equivalent (conjugate) to Eq. (106) and also to the general equation of a plane pendulum in the medium flow. Example. Let us consider the system (79), (80) under condition (15). It is also a zero mean variable dissipation system. Lemma 5.2. The infinite-dimensional space χ(Φ) of vector fields corresponding to the system (79), (80) falls into a disjoint union χ(Φ) = χ(Φ1 ) χ(Φ2 ) χ(Φ3 ) having the following properties: 808
(1) the system (79), (80) defined by using the spaces χ(Φ1 ) and χ(Φ3 ) (see Figs. 4 and 6) is relatively rough in the space χ(Φ); (2) the system (79), (80) defined by using the space χ(Φ2 ) (see Fig. 5) is a system of the first relative nonroughness degree in the space χ(Φ); (3) the set χ(Φ2 ) is of zero measure in the space χ(Φ); (4) the sets χ(Φ1 ) and χ(Φ3 ) are of finite measure in the space χ(Φ). In this case, the topological equivalence is constructed depending on the phase cylinder domain and also on the classes χ(Φk ), k = 1, 2, 3. 5.3.2. Nonzero mean variable dissipation systems in the rigid body plane dynamics. Example. In what follows (Sec. 10), we present a generic topological classification of portraits of system (92) under condition (15) for a certain infinite-dimensional parameter domain. In general, for h = 0, system (92) is a nonzero mean variable dissipation system. Lemma 5.3. The infinite-dimensional space χ(Φ) of vector fields corresponding to the system (92) for h = 0 falls into a finite disjoint union χ(Φ) = χ(Φ1 ) · · · χ(ΦN1 ) χ(Φ1 ) · · · χ(ΦN2 ) having the following properties: (1) for h = 0, system (92) defined by using the spaces χ(Φk ), k = 1, . . . , N1 , is (absolutely) rough; (2) for all k, the sets χ(Φk ) are of finite measure in the space χ(Φ); (3) the sets χ(Φk ), k = 1, . . . , N2 , are of zero measure in the space χ(Φ). Eight (but not all) “generic” classes for systems of the form (92) for h = 0 are presented in Figs. 30–37.
Fig. 30: (Absolutely) rough dynamical system.
Fig. 31: (Absolutely) rough dynamical system.
809
Fig. 32: (Absolutely) rough dynamical system.
Fig. 33: (Absolutely) rough dynamical system.
Fig. 34: (Absolutely) rough dynamical system.
Fig. 35: (Absolutely) rough dynamical system.
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Fig. 36: (Absolutely) rough dynamical system.
Fig. 37: (Absolutely) rough dynamical system. It should be noted that in all systems presented above (and in system (92), for h = 0, only in a subdomain of parameters), we can use “generic” representatives of these classes of systems, the corresponding analytic systems. Example. Let us consider the system (86), (87) under conditions (15) and (13) in the parameter domain I (see Sec. 7). Using the classes of functions Φ and Σ in which the functions F and s respectively vary, we define the space of vector fields of the system (86), (87), which is denoted by χ(B). Lemma 5.4. The infinite-dimensional space χ(B) of vector fields corresponding to a system of the form (86), (87) falls into a countable disjoint union χ(B) = χ(B1 ) χ(b1 ) χ(B2 ) χ(b2 ) . . . having the following properties: (1) a system of the form (86), (87) defined by using the spaces χ(Bi ) for any i ∈ N is (absolutely) rough; (2) the system (86), (87) defined by using the spaces χ(bi ) is a system of the first relative nonroughness degree in the space χ(B); (3) the sets χ(bi ) are of zero measure in the space χ(B); (4) the sets χ(Bi ) are of finite measure in the space χ(B). Moreover, the homeomorphism realizing the topological equivalence of systems taken from the space χ(Bi ) for each fixed i can be not sufficiently close to the identity homeomorphism. Therefore, we have the two-parameter family of phase portraits in which under passage from one portrait topological type to another, we need to deal with degenerate surgeries (Sec. 7). Moreover, the 811
subsets considered in the space J2 by the use of which the field subclasses χ(Bi ) and χ(bi ) are defined accumulate near the point (0, 0) ∈ J2 {µ1 , µ2 }. Ten classes of fields of different equivalences are depicted in Figs. 7–16. Example. Let us consider the system (86), (87) under conditions (15) and (13) in the parameters domains II and III (see Sec. 7). Using the function classes Φ and Σ in which the functions F and s respectively vary, we define the space of vector fields of the system (86), (87), which, as above, is denoted by χ(B). Lemma 5.4 holds for such vector fields. Eight classes of fields of different equivalencies are depicted in Figs. 17–24. It is seen from what was said above that in a generic case, the nonzero mean variable dissipation systems considered are (absolutely) structurally stable. Moreover, relatively structurally stable (generic) zero mean variable dissipation systems are as a rule convenient comparison systems for nonzero mean variable dissipation systems (cf. [70]). 5.3.3. Variable dissipation systems in rigid body spatial dynamics. Analogous assertions can be formulated for systems arising in the rigid body spatial dynamics. In this case, zero mean variable dissipation systems are only relatively rough, and nonzero mean variable dissipation systems are merely (absolutely) rough. The latter systems are nontrivial examples of relatively and absolutely rough systems that have order greater than 2. Because of their cumbersome character, in this section, we do not perform the qualitative analysis of such systems (of the third order and higher) (for variable dissipation systems from the spatial dynamics of a rigid body interacting with a medium, see Secs. 8 and 9). It should be noted that zero mean variable dissipation systems naturally arise in the rigid body dynamics owing to the existence of additional symmetry groups. The latter arise from natural (geometric and analytical ) problem symmetries. Therefore, along with the “utility” of appearance of zero mean variable dissipation systems as important comparison systems, they have all rights for existence because of their relative roughness. Just owing to the physical and topological nature, the nonzero mean variable dissipation systems are (absolutely) rough, which allows us to speak of the structurally stable phenomena of nature by using such systems. In the next sections, we qualitatively study nonlinear systems of the form (72); (78)–(80); (81)–(83); (85)–(87); (88)–(90), which arise in the rigid body plane dynamics. In Secs. 8 and 9, we study analogous systems from the spatial dynamics. In this case, we use the qualitative material obtained in Secs. 4 and 5. 6. Methods for Analyzing Zero Mean Variable Dissipation Dynamical Systems in Rigid Body Plane Dynamics In the first part of this section, we consider the class of motions such that a certain nonintegrable constraint is imposed on the system considered, which allows us to assume that the velocity of a certain characteristic rigid body point is constant during all the time of motion. We perform a complete qualitative analysis of the obtained dynamical system in the quasi-velocity space. We show the system symmetries and in explicit form find its first integral that is a transcendental function of quasi-velocities. Let us consider the case of body motion in which condition (79) holds during all the time of motion. As is shown in Sec. 3.3, under condition (15), system (72) is isolated in the space of quasi-velocities α and Ω. To describe the body position on the plane, we choose the Cartesian coordinates x, y of the plate center and the angle of turn φ, which is made by the line CD and the inertia system coordinate axis in the body motion plane. In this case, the phase state of the system is characterized by the tuple (x , y , φ , x, y, φ), (104) φ = Ω, x = v cos(α + φ), y = v sin(α + φ). As was mentioned, the coordinates (x, y, φ) are cyclic. Remark. Of course, in this case, the cyclicity is understood not in the Routh sense when a first integral corresponds to each cyclic coordinate, which results in reducing the order of the general Lagrangian 812
system [236–241]. Here the cyclicity is understood in the sense that the dynamical functions (kinetic energy and generalized forces) are independent of certain coordinates of the configuration space. In this case, the order of the general system is reduced by the number of such coordinates. 6.1. Case of Body Motion in a Medium under a Certain Nonintegrable Constraint and Beginning of Qualitative Analysis. 6.1.1. System equilibrium points and stationary motions corresponding to them. The set of equilibrium points of system (70) to which stationary motions correspond is partitioned into two parts. First, there exist isolated singular points, which are found from the system {α = 0 mod π, Ω = 0}. These singular points correspond to those stationary motions under which the body executes translational motion along the line CD to one or another side (see Figs. 1 and 2). Second, formally, there exists a one-dimensional manifold O of singular points of (71). A point of the singular manifold O corresponds to the stationary motion under which the body executes a rotation around a certain fixed point W with constant angular velocity; in this case, the line CD necessarily passes through the point W . The angle of attack is equal to π/2 in this case. These classes of motions slightly contradict common sense. The opposite plate borders “cut” the medium in different directions. In this case, it is necessary to take account of the existence of the medium resistance and the resistance moment. Since the introduction of the resistance moment requires account of an additional damping (see the Introduction), as was mentioned, account of such an influence is the next stage of the study. 6.1.2. Symmetries of system vector field on quasi-velocity phase cylinder. Under condition (15), the vector field of system (72) has the following two symmetry types. (1) The central symmetry. Near the points (πk, 0) and k ∈ Z such a symmetry arises owing to the fact that the system vector field in the coordinates (α, Ω) alternates its sign under the change πk − α πk + α → ; −Ω Ω (2) A certain mirror symmetry. With respect to the lines Λi , i ∈ Z (see (71)), such a system arises owing to the fact that the α-component of the system vector field in the coordinates (α, Ω) is preserved under the change π/2 + πi − α π/2 + πi + α → , Ω Ω whereas the Ω-component alternates its sign. The existence of certain mirror symmetry implies that the union of lines π (α, Ω) ∈ R2 : α = k, k ∈ Z , 2 and only it, is the isocline of the horizontal direction. The isocline of the vertical direction is only the line
σv F(α) 2 ¯ L = (α, Ω) ∈ R : Ω = . I cos α (On manifold O, we extend them by continuity.) 6.2. On Transcendental Integrability of System. This subsection is devoted to studying the possibilities of complete integrability of the dynamical system considered. Here we present a first integral of system (73) that is expressed through elementary functions and also discuss the way of integrating the general system (72) (cf. Sec. 2). 813
6.2.1. Some general assertions. The following theorem, which relates the behavior of trajectories near asymptotically limit sets and the system integrability, holds. The asymptotically limit sets are repelling and attracting limit sets. In Sec. 2, we have already dealt with the essence of proving the following theorem and present a complete list of first integrals of the corresponding system. The author considers that it is necessary to repeat this in more detail in this section. Consider a system of equations in the phase space Rn . Theorem 6.1. If the system has asymptotically limit sets, then it has no complete list of continued first integrals in the whole phase space. Although the proof of the theorem is sufficiently simple, it has an important topological meaning related to the continuity of first integrals near limit sets. Corollary 6.1. As functions, the first integrals can have essentially singular points in these asymptotically limit sets. We show below that the isolated singular points of system (70) are attracting or repelling. Therefore, if there exists a first integral of system (72) (or (73)), then it is a transcendental function [309]. 6.2.2. Transcendental system integral. Theorem 6.2. System (73) has a transcendental first integral expressed through elementary functions. Proof. Outside the manifold O, using the change τ = sin α,
y = −Ω + A1 τ
we can reduce (73) to one equation A2 τ dy =− + A1 ; dτ y by substitution ξ = τy , we separate the variables in it: (ξ 2 − A1 ξ + A2 ) dτ = −ξτ dξ. The quadrature of the latter equation yields the desired first integral (there are three cases depending on the parameters): (1) A21 − 4A2 < 0. 2A1 2ξ − A1 arctan + ln τ 2 = const; ln(ξ 2 − A1 ξ + A2 ) + 2 −A1 + 4A2 −A21 + 4A2 (2) A21 − 4A2 > 0.
2ξ − A − A2 − 4A 1 2 1 ln ln |ξ 2 − A1 ξ + A2 | + 2 + ln τ 2 = const; A1 − 4A2 2ξ − A1 + A21 − 4A2 A1
(3) A21 − 4A2 = 0.
A1 A1 |− + ln |τ | = const. 2 2ξ − A1 Passing to the variables Ω and τ , we have (1) A21 − 4A2 < 0. 2A1 2Ω + A1 τ 2 2 arctan [Ω + A1 τ Ω + A2 τ ] × exp 2 −A1 + 4A2 τ −A21 + 4A2 ln |ξ −
814
= const;
(2) A21 − 4A2 > 0. √A21 −4A2 −A1 √A21 −4A2 +A1 × 2Ω + A1 − A21 − 4A2 τ = const; 2Ω + A1 + A21 − 4A2 τ (3) A21 − 4A2 = 0.
|2Ω + A1 τ | exp −
A1 τ 2Ω + A1 τ
= const.
The theorem is proved. Corollary 6.2. For A1 = 0, the transcendental first integral transforms into the analytic first integral of the physical pendulum. 6.2.3. On expressing system first integral through elementary and nonelementary functions. To seek the system first integral, let us consider the equation 1 dy2 = − A2 g(y1 ) cos y1 + A1 g (y1 ). dy1 y2 The form of this equation is more general than the equation describing the motion of the center of mass in the classical work [458] of Zhukovskii. After the change τ = sin y1 , y1 = π2 k, k ∈ Z, cancelling by cos y1 , we have the equation 1 dg(arcsin τ ) dy2 = − A2 g(arcsin τ ) + A1 . dτ y2 dτ In particular, if condition (16) holds, then we have the case where the latter equation becomes homogeneous. As was shown above, in this case, the first integral is expressed through elementary functions. Generally speaking, this equation is equivalent to the Riccati equation whose solution is not expressed through elementary functions in the most general case. 6.3. On Mechanical Analogy with a Pendulum in Medium Flow. Let us briefly characterize the problem on the physical pendulum in the over-running medium flow, which was posed by B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov [184]. This problem allows us to detect mechanical and topological analogies. Consider a plane plate AB clamped perpendicularly to a holder OD on a cylindrical hinge O situated in the medium flow (Fig. 38). In this case, the body is the physical pendulum. The plate AB and the hinge axis are perpendicular to the motion plane. The medium flow moves with constant velocity v = 0. Assume that the holder does not create a resistance. The total force S of the medium flow action on the body is directed parallel to the holder, whereas the point N of application of this force is determined only by a single parameter, the angle of attack α, which is made by the velocity vector vD of the point D with respect to the flow and the holder OD. Such conditions arise when one uses the model of streamline flow around plane bodies. Therefore, the force S is directed along the normal to the plate to its side, which is opposite to the direction of the velocity vD , and passes through a certain point N of the plate, which is displaced from the point D forward with respect to the direction of the vector vD . The vector e = OD/σ (σ is the distance OD) determines the orientation of the holder. Then 2 e, S = s1 (α)vD
where the resistance coefficient s1 depends only on the angle of attack (cf. the results of Sec. 1). By the plate symmetry properties with respect to the point D, the function s(α) = s1 (α) sign cos α satisfies condition (13), whereas the displacement of the point N (i.e., DN = y(α)) satisfies the condition y ∈ Υ. Here Υ is the class of functions {yN } described in Sec. 1 (see (11)). It is easy to verify that the product F(α) = y(α)s(α) satisfies condition (15). 815
Fig. 38: Plane pedulum in the medium flow.
The equation of motion of the pendulum considered has the form 2 , I θ¨ = −F(α)vD
(105)
where θ is the angle of deviation of the pendulum (Fig. 38) and I is its moment of inertia. The connection between the angle of attack and the quantities θ and θ˙ are determined by the kinematic relations ˙ v = |v|. vD cos α = v cos θ, vD sin α = σ θ˙ + v sin θ, In a particular case where property (16) holds, Eq. (105) reduces to the form I∗ θ¨ + hθ˙ cos θ + sin θ cos θ = 0,
(106)
where I∗ , h > 0. Now let us return to the previous problem of the free body motion. System (73) is equivalent to the equation (107) α ¨ − A1 α˙ cos α + A2 sin α cos α = 0, where A1 , A2 > 0. Therefore, Eq. (107) is equivalent to Eq. (106) if we set α = θ + π. In this case, the change in time of the momentary angle of attack α for the free body and the angle of turn of the pendulum in the medium flow obeys the same laws [246, 247]. We can show that even if condition (16) does not hold, the obtained mechanical analogy holds for any functions F satisfying condition (15). Indeed, the equation of the physical pendulum motion in the medium flow reduces to the form ˙ = 0, I∗ θ¨ + f (α)[v2 sin θ cos υ + σvθ˙ cos θ] where the function f (α), α = arctan 816
σ θ˙ + v sin θ , v cos θ
Fig. 39: Phase portrait of the plane pendulum in the medium flow.
satisfies the conditions 0 < f∗ ≤ f (α) ≤ f ∗ < +∞ because of property (15). Therefore, we have an autonomous equation on the two-dimensional phase cylinder. By the methods developed in Secs. 4 and 5, it is easy to show that the latter differential equation and Eq. (105) are topologically equivalent. The phase portrait of Eq. (107) is depicted in Fig. 39, where instead of Ω, it is necessary to take α. ˙ The dynamical system defined by Eq. (107) is relatively structurally stable (relatively rough) with respect to the function class Φ (see Sec. 5). 6.4. Topological Structure of Phase Portrait of System Studied. In this subsection, we perform the global qualitative analysis of the dynamical system (72) under condition (15) on the whole phase plane R2 {α, Ω}. For any function F ∈ Φ, the phase portrait of system (72) has the same topological type. As follows from the analysis performed, system (72) has no trajectories having infinitely distant points of the phase plane R2 {α, Ω} as α- and ω-limit sets. In Sec. 4 (corollary from Lemma 4.6), it is shown that around the points (kπ, 0), k ∈ Z (in the strip Π, as well as in the strip Π ), there is no closed curve of trajectories for the vector field of system (72), and, in particular, there are no simple and complicated limit cycles. Because of two types of symmetries of system (72), on the phase plane, there are no closed curves of trajectories contractible to a point along the phase cylinder. 6.4.1. Topological classification of system equilibrium points. Let us perform the topological classification of singular points of system (72) under condition (15). (1) The equilibrium points (kπ, 0), k ∈ Z, of system (72) are repelling for k = 2l, l ∈ Z, and attracting for k = 2l + 1. For A21 F (0) − 4A2 < 0, the considered points are foci, for A21 F (0) − 4A2 > 0, nodes, and for A21 F (0) − 4A2 = 0, degenerate nodes. (2) On each connected component O, there exists the singular point Si of sys Λi of the manifold tem (72) with the coordinates (2i + 1) π2 , (−1)i A1 . For any i, such a point is a point of hyperbolic type, the saddle having separatrices. Because of two types of symmetries, it suffices to examine the behavior of separatrices entering (emanating from) saddles S−1 and S0 in the strip Π. The behavior of the separatrix emanating from the point S−1 to the strip Π (as well as that of the separatrix emanating from the point S0 to the strip Π) determines the global behavior of all trajectories of system (72). This separatrix is said to be key. 817
6.4.2. On key separatrix behavior. Studying the isoclines, we conclude that the key separatrix cannot go to infinity on the phase plane. Since the strip Π (and Π ; see Sec. 4) contains no closed curves of trajectories of system (72), it follows that the key separatrix either enters the point S0 or intersects the line Λ0 at a right angle. Proposition 6.1. The key separatrix intersects the line Λ0 at the point π2 , Ω∗ for Ω∗ < −A1 . The proof can be performed by using the comparison system method (see Sec. 4). 6.4.3. Phase portrait. Its relative roughness. Now let us consider the global qualitative location of trajectories of system (72) on the phase plane R2 {α, Ω}. Because of the existence ofa certain mirror symmetry for system (72) and also by Proposition 6.1, the key separatrix has the saddle − 32 π, 0 as an ω-limit set. The behavior of other symmetric separatrices can be studied analogously. Corollary 6.3. (1) The separatrix entering the point S−1 in the strip Π has the origin as an α-limit set. (2) The motions with a sufficiently large energy are periodic. (3) If we assume that the phase space of system (72) is the cylinder S1 {α mod 2π} × R1 {Ω}, then it contains a continuum of trajectories not contractible to a point. There are no closed trajectories contractible to a point along the cylinder. (4) By using the change of variables F(α) , cos α which is nonsingular on the whole phase plane, the phase portrait of the system on the plane R2 {y1 , y2 } is depicted in Fig. 39 (instead of Ω, we have y2 , and instead of α, we have −y1 ). (5) The phase portrait of system (72) is depicted in Fig. 3 (instead of α, we have −α). y1 = α,
y2 = −Ω + A1
Remark. The key separatrices are boundaries of domains in each of which the motion is of different character. So, in the oscillatory domain containing attracting and repelling equilibrium points, almost all trajectories have attractors and repellers as limit sets. Hence there is not even an absolutely continuous function that is the density of an invariant measure in this domain. The matter is different for the domain entirely filled with rotational motions. As V. V. Zhuravleva showed in her diploma work (1988), there exists a smooth function that is the density of an invariant measure in the domain entirely filled with periodic trajectories not contractible to a point along the phase cylinder. In connection with the global qualitative analysis of system (72) just carried out, let us make the following important conclusion: precisely, under condition (15), the vector fields of systems of the form (72) are topologically equivalent to each other (see Sec. 5). In particular, a system of equations of the form (72) is structurally stable with respect to the function class Φ. Generally speaking, if we consider an arbitrary deformation of the vector field of system (72), then the usual (absolute) roughness of the system violates. The latter fact is due to the existence of separatrices going from a saddle to a saddle. In Sec. 10, we show that for h = 0, the key separatrices of system (92) split. Also, in this section, for any h, under condition (15), we perform a generic topological classification of phase portraits of system (92), which describes the plane-parallel body motion in a medium under which the velocity of the plate center of mass remains constant and a damping moment linear in the angular velocity acts on the body. In what follows, we perform the qualitative analysis of the body motion trajectories on the plane. After the complete qualitative study of the quasi-velocity phase cylinder, it becomes possible to study concrete 818
rigid body trajectories. The dynamical system of the quasi-velocity space is relatively structurally stable. We perform the integration of the kinematic relations aiming at the mechanical interpretation of motion. We discuss general properties of the solution space: symmetries, various fiberings of the phase space, and its separation into the oscillatory and rotational domains. We study the properties of the solutions corresponding to the oscillatory domain: properties of asymptotes under the rigid body motion, various equivalence relations on the trajectory space, topological analogies, and mechanical interpretations of asymptotic motions. We study the properties of the solutions corresponding to the rotational domain: the existence of a family of periodic trajectories that everywhere densely fill certain domains and the problems of density of unclosed trajectories in bounded sets [282, 430]. The first results of numerical studying the body motion trajectories on the plane become a good stimulus for developing analytical tools for studying such trajectories. These numerical results show the way of developing the qualitative analysis of this dynamical system. Afterwards, when exhaustive analytical proofs are obtained for various qualitative properties of this dynamical system, the numerical construction yields quantitative characteristics of trajectories of the body motion on the plane. 6.5. General Properties of Solutions of Dynamical System. Since the topological type of the phase portrait of system (72) does not change under deformations of the vector field in the class of fields defined by condition (15), we study the following analytic dimensionless system of equations obtained from system (73): α = Ω + β sin α, Ω = −β sin α cos α.
(108)
The dimensionless kinematic relations take the form φ = Ω, x = cos(α − φ),
(109)
y = sin(α − φ). Here we have introduced the dimensionless parameter β=
σ 2 AB = σ 2 n20 > 0. I
For any β > 0, the phase portrait of system (108) is depicted in Fig. 3, and for any initial conditions (α0 , Ω0 ), we know the qualitative behavior of solutions α = α1 (t, α0 , Ω0 ), Ω = Ω1 (t, Ω0 , α0 ) of system (108). Therefore, by (109), t φ = φ1 (t, φ0 , α0 , Ω0 ) = φ0 +
Ω1 (τ, Ω0 , α0 ) dτ, t0
t x = x1 (t, x0 , α0 , Ω0 , φ0 ) = x0 +
cos α1 (τ, α0 , Ω0 ) − φ1 (τ, φ0 , α0 , Ω0 ) dτ,
t0
t y = y1 (t, y0 , α0 , Ω0 , φ0 ) = y0 +
sin α1 (τ, α0 , Ω0 ) − φ1 (τ, φ0 , α0 , Ω0 ) dτ.
t0
Since there exist the central symmetry and a certain mirror symmetry, we consider the following class of initial conditions: π π t0 = 0, α0 = , φ0 = − , x0 = y0 = 0. 2 2 819
Therefore,
π α = α1 t, , Ω0 = α2 (t, Ω0 ), 2 π = Ω2 (t, Ω0 ), Ω = Ω1 t, Ω0 , 2 t π π π φ = φ1 t, − , , Ω0 = φ2 (t, Ω0 ) = − + Ω2 (τ, Ω0 ) dτ, 2 2 2 0
π π = x2 (t, Ω0 ) = x = x1 t, 0, , Ω0 , − 2 2
t
cos α2 (τ, Ω0 ) − φ2 (τ, Ω0 ) dτ,
0
y = y1
π π t, 0, , Ω0 , − = y2 (t, Ω0 ) = 2 2
t
sin α2 (τ, Ω0 ) − φ2 (τ, Ω0 ) dτ.
0
Let us show the general properties of the solutions of the system (108), (109) corresponding to the presented class of initial conditions. Proposition 6.2. The functions α2 (t, Ω0 )− π2 , φ2 (t, Ω0 )+ π2 , and x2 (t, Ω0 ) are odd, and Ω2 (t, Ω0 ), y2 (t, Ω0 ) are even functions of time t (for any fixed Ω0 ∈ R). We can assume that Ω0 ∈ (−β, +∞) because of the existence of two types of symmetries. Corollary 6.4. On the plane R2 {x, y}, the curve F x2 (t, Ω0 ), y2 (t, Ω0 ) = 0 is symmetric with respect to the axis Oy for any Ω0 ∈ R. 6.6. Fiberings of Phase Space. 6.6.1. Two approaches to consideration of global location of trajectories in phase space. Since the independent subsystem (108) splits, the phase space admits the following fiberings. First approach. By (69), the phase space fibers into five-dimensional hyperplanes {(v, α, Ω, x, y, φ) ∈ R6 : v = const > 0}, which are bundles over two-dimensional bases, the cylinders S1 {α mod 2π} × R1 {Ω}. This family of cylinders was obtained for case (16) in explicit form since an independent first integral of the independent subsystem (73) is presented. Fix a certain level of the integral of system (73). We thus obtain a four-dimensional cylinder. Let us restrict the system of equations to this cylinder. In turn, the three-dimensional base R3 {x, y, φ} of the bundle is a bundle over the circle S1 {φ mod 2π} whose fiber is the plane R2 {x, y}, and it also has a number of properties of cylindrical nature. Second approach. By (69) and (108), the six-dimensional phase space is a bundle on the three-dimensional base R3 {v, α, Ω}, which, in turn, fibers into two-dimensional planes R2 {α, Ω}, precisely, {(v, α, Ω) ∈ R3 : v = const > 0}. On these planes, the phase portraits are the portraits of the independent system (108). Therefore, we have obtained the complete fibering of the base R3 {v, α, Ω}. By the so-called complete and explicit “integration” of the base R3 {v, α, Ω} for the bundle of the six-dimensional phase space, we perform the lift of trajectories from the base to the phase space and obtain the complete bundle of the whole phase space. 820
6.6.2. On separation of quasi-velocity phase plane into domains with different characters of trajectory behavior. The structure of the phase plane R2 {α, Ω} of system (108) completely determines the bundle of the six-dimensional phase space. Let us dwell on the analysis of phase trajectories on the plane R2 {α, Ω} in more detail. The key separatrices, i.e., the separatrices emanating from the points π − + πl, (−1)1 β 2 and entering the points π 3 + πl, (−1)1 β , l ∈ Z, 2 separate the phase plane into two domains with absolutely different characters of the trajectory behavior (see Fig. 3). The first domain is the oscillatory or finitary domain (it is simply connected and contains the line every such a trajectory {(α, Ω) ∈ R2 : Ω = 0}) entirely filled with trajectories of the following type. Almost starts from the repelling point (2πk, 0) and ends at the attracting point (2l + 1)π, 0 , l, k ∈ Z. The exceptions are the equilibrium points (πk, 0) and also the separatrices, which either emanate from the repelling points (2πk, 0) and enter the saddles S2k−1 and S2k or emanate from the saddles S2k and S2k+1 and enter the attracting points (2k + 1)π, 0 . Here π + πk, (−1)k β . Sk = 2 The second domain is the rotational domain (it is two-connected) entirely filled with rotational motions similar to rotations on the phase plane of the mathematical pendulum. These phase trajectories envelop the phase cylinder and are periodic on it. Although the dynamical system (108) is nonconservative, as was noted above, in the rotational domain of its phase plane R2 {α, Ω}, it admits the preservation of an invariant measure of variable density. This property characterizes the system considered as a variable dissipation system. 6.7. Properties of Solutions Corresponding to Oscillatory Domain. In this subsection, we perform the qualitative study of the solutions of the kinematic equations (109) corresponding to the initial conditions from the oscillatory domain of the quasi-velocity plane. 6.7.1. Properties of asymptotes under rigid body motion on the plane. We consider the functions α2 (t, Ω0 ) and Ω2 (t, Ω0 ) for t ≥ 0 and also for Ω0 ∈ (−β, Ω∗ ), where Ω∗ is such that lim Ω2 (t, Ω∗ ) = β.
t→+∞
The following relations hold for all such Ω0 : lim Ω2 (t, Ω0 ) = 0,
t→+∞
lim α2 (t, Ω0 ) = π.
t→+∞
The translational body motion under which the center of mass is ahead is exponentially orbitally stable (see [246, 247]). Therefore, on the plane R2 {x, y}, there exists a line that is an asymptote as t → +∞ for the center of mass of the rigid body plate. To characterize the properties of this asymptote, let us present an auxiliary assertion that contains the information about possible turns of the body. Proposition 6.3. (1) For any real T > 0 (T < 0), there exists an initial value Ω0 ∈ (−β, Ω∗ ) such that +∞ Ω2 (t, Ω0 ) dt > T (< T ). I1 (Ω0 ) =
(110)
0
(2) The function I1 (Ω0 ) smoothly depends on Ω0 . 821
Corollary 6.5. For any real r, there exists Ω0 ∈ (−β, Ω∗ ) such that I1 (Ω0 ) = r. Therefore, the mapping I1 : (−β, Ω∗ ) → R is surjective. By the exponential stability of the stationary motion under which the center of mass moves ahead, the limit motions for Ω0 ∈ (−β, Ω∗ ) and as t → ±∞ are asymptotic motions to lines that are symmetrically located with respect to the axis Oy: as t → +∞, to one line, and as t → −∞, to another one. Therefore, it suffices to study only one of such lines. Obviously, the angle of turn of the desired line as t → +∞ is equal to I1 (Ω0 ), i.e., it is given by integral (110). Therefore, the angular coefficient of the desired line is given by the initial value of the angular velocity Ω0 . By Proposition 6.3, there arises the (probably multivalued) mapping I2 : R → (−β, Ω∗ ). This mapping generates the mapping I3 : S1 → (−β, Ω∗ ), which is multivalued and, in some sense, is inverse to the mapping I1mod : (−β, Ω∗ ) → S1 = R/2πZ. Let us consider the mechanical meaning of the mappings I1 and I1mod , which are inverse to the mappings I2 and I3 respectively. To each admissible value Ω0 ∈ (−β, Ω∗ ), the mapping I1 puts in correspondence a real number r; taking it modulo 2π, we can obtain the angle between the asymptote and the axis Oy. Moreover, the divisor of the number r by 2π yields the signed number of turns made by the body under its motion from the initial position up to the asymptotically (as t → +∞) stable final position. To an admissible value of the initial angular velocity, the mapping I1mod only puts in correspondence the angle of turn from the axis Oy to the body motion asymptote as t → +∞, and it contains no information on the number of complete signed turns. 6.7.2. Equivalence relations on trajectory spaces. Let us consider the space of all body trajectories α2 (t, Ω0 ), Ω2 (t, Ω0 ), φ2 (t, Ω0 ), x2 (t, Ω0 ), y2 (t, Ω0 ), on the plane R2 {x, y} corresponding to solutions of the system (108), (109), i.e., the family of curves on the plane depending on the parameter Ω0 . For each fixed β > 0, let us identify all trajectories having the same value I1 (Ω0 ). Therefore, we have identified all admissible body motions under which (1) the body executes the same number of signed complete turns; (2) the body has an asymptote that makes the same angle with the axis Oy. This relation is an equivalence relation. The quotient space of the whole space of admissible trajectories by this relation is denoted by FT. Obviously, if the mapping I1 is one-to-one, then the FT coincides with the space of trajectories for which Ω0 ∈ (−β, Ω∗ ). On the circle S1 with a fixed point E1 ∈ S1 , let paths starting from the point E1 and ending at a point E2 on the circle be given. The paths from E1 to E2 on the circle are identical (homotopic) if they are contractible to one another on the circle, Such a relation is also an equivalence relation. Corollary 6.6. For E1 = E2 , we obtain the fundamental group of the circle π1 (S1 ), which is isomorphic to Z. If E1 = E2 , and, moreover, the point E2 is not fixed, then the obtained quotient group is isomorphic to π1 (S1 ) × S1 = Z × S1 . Proposition 6.4. The space FT is isomorphic to π1 (S1 ) × S1 = Z × S1 . 822
Corollary 6.7. To each admissible value Ω0 ∈ (−β, Ω∗ ), we put in correspondence the trajectory on the plane R2 {x, y}, which defines an element of the space FT, which, in turn, is given by two numbers: the integer number of complete signed turns and also the real number equal to the angle of turn of the asymptote from the axis Oy modulo 2π. Corollary 6.8. If E1 = E2 , then we obtain an isomorphism FT ∼ = π1 (S1 ). This corresponds to the following: executing several turns, the body has the asymptote as t → +∞ parallel to the axis Oy. Therefore, in this case, only one integer containing information on the number of complete signed turns corresponds to each trajectory. Corollary 6.9. If the mapping I1 (Ω0 ) is strictly monotone, then there arises a natural isomorphism between the interval (−β, Ω∗ ) and the set Z × S1 . Now, it is appropriate to make the following conclusion on the body motion trajectories on the plane: for each Ω0 ∈ (−β, Ω∗ ), the trajectory on the plane R2 {x, y} “is coded” by the pair of numbers (q, Ψ), q ∈ Z, Ψ ∈ R, Ψ mod 2π, which is the “address” of the trajectory. 6.7.3. Mechanical interpretations of asymptotic motions. Let us consider the initial values of the angular velocity Ω0 under which the asymptote is parallel to one of the coordinate axes. Case A. The asymptote is parallel to the axis Ox. There exists a sequence of points {Ωn } ⊂ (−β, Ω∗ ) such that I1mod (Ωn ) = π/2 for all n ∈ N. The number of complete turns of the body twisted with the angular velocity Ωn can attain any integer. Denote by L1 the distance from the asymptote to the line Ox. By the corollary from Proposition 6.2, the asymptote of the body motion as t → −∞ coincides with the asymptote of the body motion as t → +∞. Therefore, for any initial angular velocities Ω1 (n ∈ N), the increment of the ordinate of the body point in an infinite time (from −∞ up to +∞) is equal to zero (Fig. 40).
Fig. 40: Body motion with horizontal asymptote. Case B. The asymptote is parallel to the axis Oy. There exists a sequence of points {Ωn } ⊂ (−β, Ω∗ ) such that I1mod (Ωn ) = 0 for all n ∈ N. The number of complete turns of the body twisted with the angular velocity Ωn can attain any integer. The distance from the asymptote to the line Oy is denoted by L2 . By the corollary from Proposition 6.2, the asymptote of the body motion as t → −∞ is also parallel to the axis Oy and lies at the distance L2 from the axis Oy. Therefore, for any initial angular velocities Ωn (n ∈ N), the increment of the abscissa in an infinite time (from −∞ up to+∞) is equal to 2L2 (Fig. 41). 823
Fig. 41: Body motion with vertical asymptote.
In this case (see Corollary 6.8), there exists the following topological analogy: the space FT is isomorphic to the fundamental group π1 (S1 ). For a more detailed explanation of the latter fact, it suffices to consider all possible trajectories x(t), y(t) on the plane R2 {x, y} (and isolate the real trajectories among them as solutions of the system (108), (109)) and all possible curves on the circle S1 starting from a fixed point and ending at the circle. It is necessary to consider the latter trajectories with accuracy up to a homotopy along S1 . 6.7.4. On a certain local property of asymptote. Up to now, we considered the system (108), (109) for any fixed value of the parameter β > 0. Let us show a certain property of the asymptote of motion parallel to the axis Oy and formally depending on the parameter β > 0. Let us pose the following question: in which way the increment L2 depends on the parameter β? The quantity L2 is represented in the following form: +∞ cos αβ (t, Ω0 ) − φβ (t, Ω0 ) dt. L2 = L2 (Ω0 , β) = 0
Here αβ (t, Ω0 ), φβ (t, Ω0 ) are solutions α2 (t, Ω0 ), φ2 (t, Ω0 ) of system (108) for each β > 0. Proposition 6.5. For small β > 0 , the function L2 (Ω0 , β) is represented in the following form for any Ω0 ∈ (−β, Ω∗ ): L2 (Ω0 , β) = l2 (Ω0 , β) · β 2 , where ∂l2 (Ω0 ,β) , l2 (Ω0 , β)|β=0 are bounded. (Here the derivatives are calculated as β → +0.) ∂β
β=0
Corollary 6.10. In the case of system (108), for admissible Ω0 , the function l2 (Ω0 , β) is analytic; therefore, the representation for L2 (Ω0 , β) holds for all β ∈ R. 6.8. Properties of Solutions Corresponding to Rotational Domain. Let us consider the class of trajectories corresponding to the rotational domain. 6.8.1. Existence of a family of periodic trajectories. Let us study the general properties of periodic trajectories enveloping the phase cylinder of quasi-velocities and also the connection of these properties with the properties of the rigid body trajectories on the plane. For this purpose, we study the functions α2 (t, Ω0 ) 824
and Ω2 (t, Ω0 ) for t ≥ 0 and also for Ω0 ∈ (Ω∗ , +∞). In this case, for each fixed Ω0 ∈ (Ω∗ , +∞), Ω2 (t, Ω0 ) is a periodic function of t with period T (Ω0 ): Ω2 (t + T (Ω0 ), Ω0 ) = Ω2 (t, Ω0 ), whereas α2 (t, Ω0 ) has the increment 2π: α2 (t + T (Ω0 ), Ω0 ) = α2 (t, Ω0 ) + 2π. To begin with, we study the dependence of the trajectory period T (Ω0 ) on Ω0 ∈ (Ω∗ , +∞). Since ∂α2 (t, Ω0 ) > 0 for all t ∈ R, Ω0 ∈ (Ω∗ , +∞), ∂t along any closed trajectory on the cylinder, by the implicit function theorem, we can uniquely resolve the equation with respect to t: t = t2 (α, Ω0 ) along the periodic trajectory. Moreover, ∂t2 (α, Ω0 ) > 0 for all α ∈ R, Ω0 ∈ (Ω∗ , +∞). ∂α The function T (Ω0 ) is represented in the form 5π/2
0 < T (Ω0 ) = π/2
dα . Ω2 (t2 (α, Ω0 ), Ω0 ) + β sin α
Proposition 6.6. The function T (Ω0 ) is smooth on the interval (Ω∗ , +∞), and, moreover, lim
Ω0 →Ω∗ +0
T (Ω0 ) = +∞,
lim
Ω0 →+∞
= +0,
dT (Ω0 ) < 0 for all Ω0 ∈ (Ω∗ , +∞). dΩ0
Therefore, the function T (Ω0 ) defines a diffeomorphism T : (Ω∗ , +∞) → (0, +∞). Now let us consider the class of functions x2 (t, Ω0 ), y2 (t, Ω0 ) for Ω0 ∈ (Ω∗ , +∞). By Proposition 6.2, x2 (t, Ω0 ) is an odd function of t and y2 (t, Ω0 ) is an even function of t. By (109), d x2 (t, Ω0 ) = cos α2 (t, Ω0 ) − φ2 (t, Ω0 ) , dt d y2 (t, Ω0 ) = sin α2 (t, Ω0 ) − φ2 (t, Ω0 ) . dt Fix Ω0 . The increment of the function φ2 (t, Ω0 ) for time T (Ω0 ) is equal to π ∆φ = φ2 (T (Ω0 ), Ω0 ) − φ2 (0, Ω0 ) = φ2 (T (Ω0 ), Ω0 ) + > 0, 2 since Ω2 (t, Ω0 ) > 0 for all t ∈ R, Ω0 ∈ (Ω∗ , +∞). If ∆α = α2 (T (Ω0 ), Ω0 ) − α2 (0, Ω0 ), then obviously, ∆α = 2π. Let us consider the quantity ∆(α − φ) = α2 (T (Ω0 ), Ω0 ) − φ2 (T (Ω0 ), Ω0 ) − α2 (0, Ω0 ) + φ2 (0, Ω0 ). By Propositions 6.3 and 6.6, the function ∆(α−φ)(Ω0 ) is smooth for Ω0 ∈ (Ω∗ , +∞). Denote ∆(α−φ)(Ω0 ) by ∆α−φ (Ω0 ). The following inequality holds: −∞ < ∆α−φ (Ω0 ) < 2π. In other words, the increment of the difference of angles during the period can be sufficiently large. 825
Proposition 6.7. For any sufficiently large negative D < 0, there exists Ω0 ∈ (Ω∗ , +∞) such that ∆α−φ (Ω0 ) = D. Let us consider the values of the right-hand sides of the latter relations on the period T (Ω0 ), i.e., for t = T (Ω0 ): d d x2 (T (Ω0 ), Ω0 ) = cos z, y2 (T (Ω0 ), Ω0 ) = sin z. dt dt In other words, x2 (t, Ω0 ) = cos f (t) and y2 (t, Ω0 ) = sin f (t), where f T (Ω0 ) = z. Proposition 6.8. The functions cos f (t) and sin f (t) are periodic if and only if z and π are rationally dependent. Proposition 6.9. On the numerical ray (Ω∗ , +∞), there exists an everywhere dense countable set K having the following property: Ω0 ∈ K if and only if the functions x2 (t, Ω0 ) and y2 (t, Ω0 ) are periodic of period τ (Ω0 ). Since the functions x2 (t, Ω0 ) and y2 (t, Ω0 ) are periodic of period τ (Ω0 ) on the set K (for a fixed Ω0 ∈ K), it follows that the functions x2 (t, Ω0 ) and y2 (t, Ω0 ) are periodic if and only if τ (Ω0 )
x2 (t, Ω0 ) dt
0
τ (Ω0 )
y2 (t, Ω0 ) dt = 0.
=
(111)
0
Relation (111) holds on the set K since the function y2 (t, Ω0 ) is odd in the variable t and the following property also holds: π d d y2 t + , Ω0 = x2 (t, Ω0 ) for all Ω0 ∈ K. dt 2 dt Therefore, the following proposition holds. Proposition 6.10. The curve γ(t) : {(x, y) ∈ R2 : x = x2 (t, Ω0 ), y = y2 (t, Ω0 )} on the plane R2 {x, y} is closed if and only if Ω0 ∈ K. Corollary 6.11. For almost all Ω0 ∈ (Ω∗ , +∞), the trajectory of motion for the center of mass of the rigid body plate on the plane is an unclosed curve. Corollary 6.12. Since property (65) holds, it follows that either the curve goes to infinity or there exists a set in which it is everywhere dense. 6.8.2. On density of unclosed trajectories. In the system considered, there exists an infinite everywhere dense family of closed trajectories. Moreover, unclosed trajectories are everywhere dense. To study this problem, let us turn to Theorem 4.9 from which important consequences are obtained. Proposition 6.11. The unclosed trajectory, which is the projection of an integral trajectory of system (108), (109) on the plane R2 {x, y}, is everywhere dense near itself. We can show that in our case, the trajectory of the rigid body center of mass fills a bounded domain, an annulus on the plane. In Figs. 42 and 43, we show the trajectories of the plate center for small and large times respectively. This visualization of motion is obtained by using a personal computer with a specially written program package. Therefore, for almost all values of the initial angular velocity, the trajectory of the center of mass of the plate is everywhere dense in a bounded set of the plane, and it is also everywhere dense near itself. For other values of the initial angular velocity, the desired curve is closed and is given by functions of long period. 826
Fig. 42: Everywhere dense trajectory in the annulus for small times.
Fig. 43: Everywhere dense trajectory in the annulus for large times.
6.9. On Technical Tools for Model Study. In this subsection, the author briefly describes applied program package, which allows him to visualize the body motion on the plane. Since the analytical study of the behavior of the system “as a whole” is very difficult, a detailed numerical modelling of the system motion with various parameter values was carried out. For this purpose, a program that allows one to design the mechanical model sequentially and to study its motion in the desired parameter range was created. The program work process is partitioned into several stages: the computer design of the mechanical model, the choice of the initial conditions, and the modelling of motion. In this connection, it is interesting to describe the technical tools for the model study. (1) Computer design of the mechanical model. At the first stage of the program work, on the computer screen, there arises a list of the system mechanical parameters, the whole screen picture of the mechanical model. Then the desired change of parameters of the model described that control the model properties is performed. Therefore, to obtain the required configuration and the equations of motion, one performs a sequential design of the mechanical model, which is used for subsequent calculations. (2) Choice of the initial conditions. The next step forewarning the direct modelling of the motion is the assignment of the initial conditions. The configurational plane of generalized coordinates is shown on the screen, and the assignment of the initial conditions is performed by choosing a point on this plane or by inputting the exact values of the initial conditions. 827
(3) Modelling of motion. In the process of system behavior modelling, the equations of motion are integrated under the chosen initial conditions, and, moreover, the results of computations are directly outputted on the screen in the integration process. The output of results is performed in two graphical regimes. In the first regime, by using rapid computer graphic tools, the phase portrait of the dynamical system is permanently constructed in the integration process. This regime serves for detecting and analyzing characteristic types of motions on the phase plane: equilibrium states, closed trajectories, etc. In the second regime, in parallel with the computations, the real picture of motion of the system studied that executes translational-rotational motions in the medium flow is demonstrated on the screen. By introducing a number of simplifying assumptions, we have carried out the reduction of the order in a certain problem of medelling the plane-parallel body motion in a medium under streamline flow-around or separate flow. The reduced system admits a complete qualitative analysis on the quasi-velocity phase plane and a geometric interpretation of motion. We have shown that the trajectory of the plate center on the plane has axial symmetry, and under certain initial conditions, it has spherical symmetry. The phase space of the problem has a number of properties that characterize the existence of bundles. The key separatrices partition the quasi-velocity plane into two domains, the oscillatory (finitary) domain and the rotational domain. The orbital exponential stability of motion from the oscillatory domain allows us to speak of the existence of asymptotes for trajectories of motion. For any angle given in advance, there exist initial conditions of motion under which the body turns by this angle in an infinite time. Therefore, the motion can have any asymptote. Owing to a number of geometric and topological analogies, we have shown that in trajectory space, it is possible to introduce equivalence relations. Moreover, each trajectory “is coded” by an integer number of complete turns and a real number of asymptote turns modulo 2π. Depending on the unique dimensionless parameter, the translation body increment in an infinite time is proportional to the square of this parameter. The body motions corresponding to the rotational domain of the quasi-velocity plane can be either periodic or everywhere dense in a bounded domain of the plane. The period of motion along such a trajectory can be arbitrarily large. Moreover, almost all trajectories (in the measure-theoretic sense) are everywhere dense. There exists an infinite everywhere dense family of closed trajectories in the phase space. Aperiodic trajectories are everywhere dense in the spherically-symmetric domain and are also dense near themselves. Therefore, the motions in a bounded domain correspond to trajectories from the rotational domain of the quasi-velocity phase space as projections of the configurational space, and motions going to infinity correspond to trajectories from the oscillatory (finitary) domain. As is known, the study of the dynamics of a rigid body with constant velocity of the center of mass is equivalent to the study of the dynamics of motion around the center of mass in the K¨ onig coordinate system. In accordance with this, in what follows, we perform the global qualitative analysis of system of differential equations arising in the description of the rigid body motion in a resisting medium under which only a pair of forces acts on the body. In this case, the study is performed not in the K¨ onig coordinate system but in an inertial coordinate system different from it since the systems of equations considered are convenient comparison systems for more complicated (nonzero mean variable dissipation) systems, which are considered later. However, in what follows, we show three different topological types of system phase portraits and also the transcendental integrability. 6.10. Reduction of System to a Physical Pendulum. 6.10.1. On reduction of a dynamical system to a physical pendulum. In the consideration of a given class of motions described by the system (8)–(10), (104) under condition (76), we can proceed in two ways. 828
First, as we already said, in the inertial K¨ onig coordinate system, only the group of two equations is nontrivial, more precisely, φ = −Ω, Ω = F(α)v2 ; it is necessary to add two kinematic relations connecting the quantities α, Ω, and vD , where D is the center of the rigid body plate center, to this group. In this case, we are necessarily under the conditions of Sec. 6.3 whose dynamical system is equivalent to the latter system. Second, from the dynamical system theory viewpoint, considering the system (8)–(10), (104), we conclude that it is integrable. In general, for this purpose, it suffices to have five first integrals. But since the system (8)–(10) is an independent third-order subsystem in the system(8)–(10), (104), it suffices to integrate this subsystem and then to substitute the obtained functions in system (104). In this case, we qualitatively find the quantities φ, x, and y. Since the center of mass moves rectilinearly and uniformly, the system (8)–(10), (104) has two independent first integrals v cos(α + φ) − σΩ sin φ = C1 , v sin(α + φ) + σΩ cos φ = C2 ,
C1 , C2 = const.
The left-hand sides of the latter relations are components of the velocity vector for the center of mass in the immovable coordinate system. An additional first integral independent with the latter two integrals will be obtained below. Therefore, it suffices to know not five but three independent first integrals. Indeed, having three independent relations for four variables φ, v, α, and Ω, we can express the latter three quantities through φ at each point of the phase space. By the first of the kinematic relations (104), the quantity φ is found by using quadratures. Hence we find v, α, and Ω. After that, it is easy to transform two kinematic relations (104) into quadratures of the coordinates {x, y}. On the other hand, since the velocity of the center of mass is at least constant, system (8)–(10) has an analytic first integral, which, together with the additional first integral, also allows us to integrate the complete sixth-order system. 6.10.2. Existence of analytical integral. By (76), the velocity of the center of mass is a first integral of the system (8)–(10), precisely, the function Φ0 (v, α, Ω) = v2 + σ 2 Ω2 − 2σΩv sin α = vc2
(112)
of phase variables is constant on phase trajectories. Under a nondegenerate change of the independent variable, the system (77)–(80) also has an analytical first integral, precisely, the function Φ1 (v, α, ω) = v2 (1 + σ 2 ω 2 − 2σω sin α) = vc2
(113)
of phase variables is constant on phase trajectories. Relation (113) allows us to find the dependence of the center of mass of the plate on other phase variables without solving the system (78)–(80); precisely, for vc = 0, v2 = vc2 (1 + σ 2 ω 2 − 2σωv sin α)−1 . Since the phase space R1+ {v} × S1 {α mod 2π} × R1 {ω} of the system (78)–(80) is three-dimensional, and as we show below, there exist asymptotic limit sets in it, relation (113) defines a unique analytical (even continuous) first integral of the system (78)–(80) on the whole phase space. Analogously, the first integral (112) is the last analytical (even continuous) first integral of the system (8)–(10) on the whole space. Below we examine the problem of existence of the second (additional) first integral of the system (78)–(80). The phase space of the system (78)–(80) is fibered into surfaces {vc = const} similar to that in the case where v ≡ const. 829
6.11. Beginning of Qualitative Analysis. Equilibrium Points of Systems and Stationary Motions. The third-order system (8)–(10) has equilibrium points that fill one-dimensional manifolds. Since the system (78)–(80) reduces to a second-order subsystem, in the phase space of the latter, the equilibrium points can be the projections of the whole phase trajectories of the three-dimensional phase space. Therefore, the problem on equilibrium points falls into the following two problems: the problem on equilibrium points of the system (78)–(80) in the phase space R1+ {v} × R2 {α, ω} and the problem on equilibrium points of the truncated system (79), (80) on the two-dimensional phase cylinder S1 {α mod 2π} × R1 {ω} or on the plane R2 {α, ω}. 6.11.1. Equilibrium points of third-order system. The equilibrium points of the three-dimensional space are given by the systems α = 2πk, k ∈ Z,
ω = 0,
v = v1 ,
(114)
(115) α = (2k + 1)π, k ∈ Z, ω = 0, v = v2 , π (116) α = + πl, l ∈ Z, ω = 0, v = v3 , 2 1 π (117) α = − + 2πl, l ∈ Z, ω = − , σ = 0, v = v4 , 2 σ 1 π (118) α = + 2πl, l ∈ Z, ω = , σ = 0, v = v5 . 2 σ Here {vi }5i=1 is a tuple of positive constants. In the phase space R1+ {v} × R2 {α, ω}, systems (114)–(118)) define one-dimensional manifolds (lines) entirely filled with equilibrium points of the system (78)–(80). The mechanical interpretation of the equilibrium points given by systems (117), (118) coincides with the mechanical interpretation of equilibrium points of system (68), and, in this case, W ≡ C; the mechanical interpretation of the equilibrium points given by systems (114), (115) was also considered above. The equilibrium points given by system (116) are interpreted as follows: the body translational moves parallel to the plate. 6.11.2. Equilibrium points of second-order system. Let us consider the truncated system (79), (80). It has the equilibrium points on the phase plane R2 {α, ω}, which are given by the relations α = 2πk, k ∈ Z,
ω = 0,
(119)
α = (2k + 1)π, k ∈ Z, ω = 0, (120) π (121) α = + πl, l ∈ Z, ω = 0, 2 1 π (122) α = − + 2πl, l ∈ Z, ω = − , σ = 0, 2 σ 1 π (123) α = + 2πl, l ∈ Z, ω = , σ = 0. 2 σ There are no nonsingular phase trajectories that are projected into the points in the plane, which are given by systems (119)–(123). Proposition 6.12. In the phase space R1+ {v} × R2 {α, ω}, for any function F satisfying condition (15), the systems of the form (78)–(80) have no particular solutions of the form α = α1 (q) ≡ α0 = const, ω = ω1 (q) ≡ ω 0 = const, v = v1 (q) ≡ v0 = const. Proposition 6.12 immediately implies Proposition 6.13. 830
Proposition 6.13. Under condition (15), all equilibrium points of the system (79), (80) lie on the lines {(α, ω) ∈ R2 : sin 2α = 0}. In other words, the equilibrium points given by systems (119)–(123) exhaust the whole set of equilibrium points of systems having the form (79), (80). Remark. Since the analytical first integral (113) is independent of the choice of F, Proposition 6.13 holds for all systems of the form (79), (80) under condition (15). Remark. Since the isoclines of the vertical and horizontal directions transversally intersect at the equilibrium points, we can show that for any sufficiently small deformation of the vector field of (79), (80), Propositions 6.12 and 6.13 remain valid. 6.12. Bundles of Phase Space, Its Symmetry, and Beginning of Topological Analysis. 6.12.1. Topological classification of singular points of truncated system. We begin the topological analysis of the system studied with the classification of singular points. In what follows, as previously, F (0) . I The equilibrium points given by system (119) are repelling: for σn0 < 2, they are unstable foci, for σn0 = 2, unstable degenerate nodes, and for σn0 > 2, unstable nodes. The equilibrium points given by system (120) are attracting: for σn0 < 2, they are stable foci, for σn0 = 2, stable degenerate nodes, and for σn0 > 2, stable nodes. The equilibrium points given by system (121) are saddles. The equilibrium points given by systems (122) and (123) are not hyperbolic since the eigenvalues of the matrix of the linearized system near them lie on the imaginary axis. Therefore, the vector field linearized near points (122), (123) does not allow us to speak of the stability character of these point. We have the problem of distinguishing the center and the focus [13]. n20 =
(1) (2) (3) (4)
Proposition 6.14. For the system of equations (79), (80) considered under condition (15), the topological type of the singular points given by systems (122) and (123) is the center. In what follows, we explicitly obtain a first integral of systems of the form (82), (83). Therefore, we obtain an additional first integral for the system (81)–(83). 6.12.2. Bundles of phase space and its symmetries. The phase trajectories of the system (78)–(80) lie on surfaces that fiber the three-dimensional phase space and are two-dimensional cylinders. In particular, if an additional phase integral of the system (78)–(80) exists on the whole phase space, then it is a function of the variables (α, ω), and, therefore, defines the family of cylinders in R1+ {v} × R2 {α, ω}. Remark. To solve the problem on lifting the trajectories to the three-dimensional phase space completely, it is necessary to reveal the sign of the projection of the vector field on the axis p in the space R3 {p, α, ω} or on the axis v in the domain {(v, α, ω) ∈ R3 : v > 0}. Let us consider the surface M, which is the union ¯ M = M1 ∪ M, of surfaces, where
1 3 M1 = (p, α, ω) ∈ R : ω = F(α) , I ¯ and, in turn, M is represented in the form π ¯ = Mi , Mi = (p, α, ω) ∈ R3 : α = + πi . M 2 i∈Z
831
The listed surfaces are cylinders that partition the phase space into domains in each of which the projection of the vector field on the axis p has a definite sign. On the surface M itself (and only on it), the projection of the field on the axis p vanishes. Remark. For vector fields of systems having the form (79), (80), the central symmetry holds under condition (15). The vector field of the system (79), (80) (as well as the vector fields of system (72)) has a certain mirror symmetry. It is that under the change of variables −π/2 − α + πk −π/2 + α + πk → , k ∈ Z, ω ω the ω-component of the field alternates the sign, whereas the α-component remains unchanged. 6.13. On Existence of an Additional Transcendental Integral. System (78)–(80) has the analytical integral (113). This is a unique analytical (even continuous) first integral on the whole phase space. If we continue the additional integral to the complex domain, then the points that are stable or unstable foci (nodes) become essentially singular for it. Indeed, at these points, the additional integral has no limit as a function. Moreover, it is a transcendental function and is almost never expressed through elementary functions. The system (81)–(83) is a particular representative of the class of systems having the form (78)–(80), but the problem of expressing the additional integral through elementary functions is not trivial for it [225, 247, 260]. The further material was partially presented in Sec. 2. Despite this circumstance, the author considers it reasonable to present this material in more detail. To the system (82), (83), we put in correspondence the differential equation −n20 τ + σω[ω 2 − n20 τ 2 ] dω , = dτ ω + σn20 τ + στ [ω 2 − n20 τ 2 ]
τ = − sin α.
Introduce the following notation: C1 = 2 − σn0 , C2 = σn0 , and C3 = −2 − σn0 . Performing a number of changes of variables by the formulas ω − n0 τ = u1 ,
ω + n0 τ = v1 ,
u1 = v1 t1 ,
v12 = p1 ,
where v1 = 0, we obtain the Bernoulli-type equation
dp1 2σ 2p1 C1 t1 + C2 + t 1 p1 = [C3 − C1 t21 ]. n0 dt1 By the known change p−1 = q1 for p1 = 0, we reduce the latter equation to the form q1 = a1 (t1 )q1 + a2 (t1 ), where
2(C1 t1 + C2 ) 4σt1 . , a2 (t1 ) = 2 C1 t1 − C3 n0 (C1 t21 − C − 3) (Here the dot denotes the derivative in t1 .) The general solution of the linear homogeneous equation is represented in the form q1 hom (t1 ) = k(C1 t21 − C3 )Q(t1 ), k = const, where the function Q has the following form depending on the value of the constant C1 : t C1 = 0, e 1, C C 2√ 2 arctan − C1 t1 3 , C1 > 0, Q(t1 ) = e −C1 C3 C2 √ √ √ C1 C3 √−C1 t1 +√−C3 , C1 < 0. −C t − −C a1 (t1 ) =
1 1
832
3
To obtain the solution of the inhomogeneous equation, assume that the quantity k is a function of t1 , which is found by the quadrature 4σ t1 Q−1 (t1 ) dt1 . k(t1 ) = 2 n0 (C1 t1 − C3 )2 Therefore, the additional integral of the system (82), (83) takes the form −1
Q
(t1 )q1 (C1 t21
−1
− C3 )
4σ − n0
t1
Q−1 (τ1 )
t0
τ1 dτ1 = C, − C3 )2
(C1 τ12
where C = const. As is seen, the final form of the first integral depends on the sign of the constant C1 ; as a result, three variants are possible. Let us examine each of them. First variant. C1 = 0. After an elementary calculation, we obtain the additional integral in the form u σ 2 u1 − v1 −2 1 v1 + +1 = const. (124) e 2 v1 Therefore, for C1 = 0, the first integral (124) of the system (82), (83) is expressed through elementary functions. Let us prove that this integral can be obtained from the first integral of system (73) for A21 − 4A2 = 0. Indeed, passing to the variables {α, Ω} in relation (124), for σn0 = 2, we have
2 2 σ Ω + 2σΩv sin α + v2 Ω − n0 v sin α = const. (125) exp − Ω + n0 v sin α (Ω + n0 v sin α)2 By (112), the numerator standing in the square brackets is a constant function on the phase trajectories of the system (78)–(80). Therefore, considering the power (−2) of (125), we conclude that the following relation once again holds on the phase trajectories of the system (78)–(80):
1 Ω − n0 v sin α exp (126) [Ω + n0 v sin α] = const. 2 Ω + n0 v sin α Multiplying (126) by e3/2 , we do not violate the latter relation on the phase trajectories studied. In this case the exponent of the exponential takes the form 2Ω + n0 v sin α . Ω + n0 v sin α On the other hand, considering the K¨ onig coordinate system in which the axis Cx is chosen along the velocity vector of the center of mass vc , we have the kinematic relations v cos α = −Vc cos φ,
−v sin α = Vc sin φ − σΩ,
which are written in the initial phase coordinates v, α, Ω, φ. In such a K¨ onig coordinate system, the equation of motion of the obtained physical pendulum placed in the medium flow is equivalent to the system φ = y + 2n0 Vc sin φ, y = −n20 Vc2 sin φ cos φ (127) (here, we take into account that σn0 = 2). By the existence of the additional first integral of system (73) for A21 − 4A2 = 0, system (127) has a first integral of the form
n0 Vc sin φ [Ω + n0 Vc sin φ] = const. exp Ω + n0 Vc sin φ 833
This relation holds on the phase trajectories of system (127) and is equivalent to the relation
2Ω + n0 v sin α [Ω + n0 v sin α] = const, exp Ω + n0 v sin α which coincides with relation (126). The following theorem holds. Theorem 6.3. The first integral of the dynamical system describing the physical pendulum is constant on the phase trajectories of the system (82), (83). In a similar way, the theorem can also be proved for the case σn0 = 2. Second variant. C1 > 0. The integration leads to the function C2 ζ C σ −2 √−C 1 C3 √ 2 e sin 2ζ + cos 2ζ + const, − 4n0 −C1 C3 where
C1 t1 . C3 As is seen, in the case C1 > 0, the additional first integral is expressed through elementary functions (see also [135]). ζ = arctan
−
Third variant. C1 < 0. By equivalent transformations, the integral transforms to the form 1−γ ζ σ ζ −γ ζ −1−γ 2 −3 + + const, C1 C2 n0 γ−1 γ γ+1 where
√ √ −C1 t1 + −C3 C2 √ > 1, ζ = √ . γ=√ C1 C3 −C1 t1 − −C3 Therefore, in the case C1 < 0, the additional integral is expressed through elementary functions. Therefore, the following corollaries of Theorem 6.3 hold. Corollary 6.13. The topological type of equilibrium points of the system (82), (83) given by relations (122) and (123) is the center. Corollary 6.14. The system (82), (83) on the phase cylinder R1 {ω} × S1 {α mod 2π} has no simple and complicated limit cycles. In conclusion of this subsection, we note the indisputable fact that the explicit finding of the first integral of the system (82), (83) allows us to study the qualitative structure of partition into trajectories of the system on the phase cylinder S1 {α mod 2π} × R1 {ω}. Despite this circumstance, such a study can be performed by qualitative methods, which are proposed in the next subsection (see also [2, 8]). 6.14. Topological Structure of Phase Portraits of System on a Two-Dimensional Cylinder. 6.14.1. Introduction to topological classification of phase portraits. Previously, we have carried out the topological classification of equilibrium points of the system (79), (80). In Sec. 4.6 (Lemma 4.12), we have proved the existence and uniqueness of trajectories going to infinity [264]. The infinitely distant points (kπ, ±∞), k ∈ Z, and only they, are α- and ω-limit sets of such trajectories. In Sec. 4.2, using the Poincar´e topographical system method, we have shown the absence of closed curves of trajectories in the strip Π, as well as in the strip Π (Lemma 4.7). Therefore, around the points (πk, 0), k ∈ Z, there are no closed curves of trajectories. By the 2π-periodicity of the phase portrait and 834
also by its central symmetry, the closed curves of trajectories can simultaneously exist only around the points π π π 1 π 1 ,0 , − ,0 , , , − ,− (0, 0), 2 2 2 σ 2 σ the sum of whose indices is equal to 1. Since the system (79), (80) has a certain mirror symmetry, the latter is impossible. Proving Lemma 4.7 by the PTS method (see Sec. 4.2), we reveal that the separatrix entering the saddle π2 , 0 has the origin as its α-limit set. By the existence of two types of symmetries, the behavior of other analogous separatrices is similar to that just described, and the key problem of the πtopological classification is the problem of the behavior of the separatrix emanating from the point − 2 , 0 to the strip Π. Such a separatrix is said to be key. 2 the points (2πl, 0), 6.14.2. On limit sets of key separatrix. As was already noted, on the plane R {α,ω}, π l ∈ Z, are repelling, whereas the points (2l + 1)π, 0 are attracting. The points 2 + πl, 0 are saddles, and, therefore, they have asymptotic curves, the separatrices; each saddle has two separatrices entering it and two separatrices emanating from it. On the lines π Λi = (α, ω) ∈ R2 : α = + πi , i ∈ Z, 2
the topological type of equilibrium points with nonzero ordinate is the center [262]. The construction of the phase portrait of the system (79), (80) is based on the study of the global qualitative behavior of the key separatrix. Studying the isoclines, we can assert that the following three variants are possible: the desired separatrix has the infinitely distant point (−0, −∞) as its ω-limit set; it transversally intersects the line {(α, ω) ∈ R2 : α = 0} and then goes to the strip Π; enveloping the equilibrium point that is the center, it returns to the point (− π2 , 0). Let us show that all three variants are possible under certain conditions [247, 260]. Introduce the general family of strips Π(α1 ,α2 ) = {(α, ω) ∈ R2 : α1 < α < α2 }; the strips introduced previously have the form Π(−π/2,π/2) = Π and Π(π/2,3π/2) = Π . Lemma 6.1. Fix a value n0 > 0. Consider the separatrices of the system (79), (80) emanating from the point − π2 , 0 to the strip Π for two different values σ = σ1 and σ = σ2 (σ1 < σ2 ). Then the inequality Σ2 (α) < Σ1 (α) holds at those points at which these separatrices corresponding to the values σ1 and σ2 in the strip Π can simultaneously be given in the form of the graphs ω = Σi (α),
i = 1, 2.
Lemma 6.2. For a certain dimensionless value (σn0 )∗ , in the strip Π(−π,0) , let the following homoclinic situation hold for the system (79), (80): the separatrix emanating from the point − π2 , 0 to the strip Π enters the same point. Then for any σn0 > (σn0 )∗ , such a homoclinic situation is preserved for the system (79), (80). ¯ such that for any σ < σ ¯ , the separatrix Lemma 6.3. For any fixed n0 , there exists a sufficiently π small σ the line of the system (79), (80) ! " emanating from the point − 2 , 0 to the strip Π transversally intersects (α, ω) ∈ R2 : α = π2 at the point (α, ω ∗ ) for a certain ω ∗ > 0 and enters the saddle 32 π, 0 . Lemma 6.4. For any function F ∈ Φ, there exists σ ∗ such that for any σ > σ ∗ , the homoclinic situation holds for systems of the form (79), (80) in the strip Π(0,π) . 835
Therefore, for any fixed function F ∈ Φ and for certain values π of σn0 , all three possible cases of for a behavior are realized for the separatrix emanating from the point − 2 , 0 to the strip Π. Moreover, fixed n0 and for small σ less than a certain σ ∗ , the desired separatrix enters the point π2 , 0 , for σ = σ ∗ , it goes to infinity, and for σ > σ ∗ , the homoclinic situation holds, which is mentioned in Lemma 6.2. Note that by the strict monotonicity property, the value σ ∗ is unique, and the mentioned surgeries of the ω-limit set are executed monotonically. For more general assertions, see Sec. 4. 6.14.3. Topological classification of phase portraits of system. Remarks on relative structural stability. The phase portrait of the system (79), (80) is realized in three different topological types. from the point − π2 , 0 Case 1. Let the point 32 π, 0 be the ω-limit set of the separatrix π emanating to strip Π. Then the separatrix emanating from the point 2 , 0 to the strip Π enters the point the 3 − 2 π, 0 . Therefore, there exists a simply connected domain on the phase cylinder whose boundary is the separatrices just mentioned such that inside it almost all nonsingular trajectorices emanate from repelling points and enter attracting points. Outside the mentioned domain, the picture of partition into trajectories is as follows. Because of the existence and uniqueness of the trajectory going to infinity, the trajectory whose α-limit set is the point (−π + 0, −∞) has the point (−0, −∞) as its ω-limit set and also bounds a domain containing a continuum of closed trajectories around the point − π2 , − σ1 . Because of the regularity of the system vector field, in other domain, there exist closed trajectories not contractible to a point along the phase cylinder. The phase portrait for this case is depicted in Fig. 4. Case 2. Let emanating separatrix the infinitely distant point (−0, −∞) be the ω-limit set of the from the point − π2 , 0 to the strip Π. Then the trajectory entering the point π2 , 0 emanates from the infinitely distant point (+0, +∞) in the strip Π(0,π) . The latter two separatrices bound a domain entirely filled with nonisolated closed trajectories that contain the singular point π2 , σ1 inside them. Almost all other trajectories emanate from repelling points and enter attracting points. The phase portrait for this case is depicted in Fig. 5. Case 3. In the strip Π(−π,0) , let the homoclinic situation mentioned above be realized (Lemma 6.4). Then there exist unique trajectories in the corresponding strips that emanate from (enter) repelling points (attracting points) and having infinitely distant points as their ω- (α-) limit sets. In the other domain, trajectories of the system vector field are completed in a way similar to Cases 1 and 2. The phase portrait for this case is depicted in Fig. 6. Under condition (15), systems of the form (79), (80) are relatively structurally stable in the following sense. Taking account of parameters σ and n0 , the topological equivalence of portraits is constructed depending on the portrait type itself. The structural stability is understood with respect to the function class Φ. 6.14.4. Example: an estimate for dimensionless parameter for which there arises a surgery of topological types of system phase portraits. To estimate the constant (σn0 )∗ (see the conclusion after Lemma 6.4) for the system of the form (82), (83), let us study the behavior of the key separatrix emanating particular from the point − π2 , 0 to the strip Π. By the monotone dependence of the vector field of the system (82), (83) on the parameter σ, the following theorem holds. Theorem 6.4. The constant value (σn0 )∗ under which there arises the surgery of the topological type of the phase portrait for the system (82), (83) satisfies the inequality (σn0 )∗ < 2. Corollary 6.15. The phase portrait of the system (82), (83) for σn0 ≥ 2 is depicted in Fig. 6. Therefore, already for a certain value σn0 < 2, the third possible phase portrait for the system (82), (83) is realized (Fig. 6). 836
6.15. Mechanical Interpretation of Some Singular Phase Trajectories. In the phase portraits (see Figs. 4–6), for certain initial conditions, there exists a singular phase trajectory which goes to infinity as the independent parameter q tends to a certain value q0 . For example, in the domain Π∩{(α, ω) ∈ R2 : ω > 0}, there exists a trajectory tending to an infinitely distant point (0, +∞) as q → q0 . Let us make the converse passage from the independent parameter q to time t. Since Ω = ωv, in the domain {(α, ω) ∈ R2 : ω > 0} ∩ Π, where the trajectory goes to infinity, the projection of the system vector field on the axis v is negative. Proposition 6.15. The trajectory going to infinity in the space R1+ {v} × S1 {α mod 2π} × R1 {ω} and parametrized by the natural parameter q is a closed trajectory in the space R1+ {v}×S1 {α mod 2π}×R1 {Ω}. The discontinuity of the trajectory studied in the space R1+ {v} × S1 {α mod 2π} × R1 {Ω} for t = t0 is due to the fact that the Jacobian of the change of the independent variable q vanishes for t = t0 . Therefore, on the plane R2 {x, y}, the cycloid with cusp is the trajectory of the plate center under the initial conditions considered. As for mechanical interpretation of other phase trajectories, it can be performed not by the method for integrating kinematic relations but either by studying the level surfaces of the first integral of the system or by a qualitative integration and interpretation of trajectories on the phase cylinder S2 {α mod 2π}×R1 {Ω} (see Fig. 3). The latter trajectories are easily integrated since they describe the motion of the physical pendulum in the medium flow. It remains only to add the translational velocity vc of the rigid body motion and obtain an explicit picture of velocity distribution in the body under the absolute motion. This section presents the qualitative material for studying the most interesting case of motion from the applied viewpoint, the rigid body free drag in a resisting medium. The next section highlights many dynamical effects of such a motion on the basis of the qualitative results obtained in this section and is in fact an introduction to the free drag problem. 7. Methods for Analyzing Nonzero Mean Variable Dissipation Dynamical Systems in Plane Dynamics of a Rigid Body We perform the most interesting (from the applied viewpoint) class of rigid body motion, a free drag in a resisting medium. This material is in fact an introduction to nonlinear problem of a free drag. We obtain some particular solutions of the complete system and present the material for performing the qualitative integration of dynamical equations in the quasi-velocity space. We obtain a new two-parametric family of phase portraits on a two-dimensional cylinder. We show that the obtained family consists of an infinite set of phase portraits with different qualitative properties. 7.1. Beginning of Qualitative Analysis. Equilibrium Points of Second- and Third-Order Systems. In the phase space R2 {α, ω} of the system (86), (87), the equilibrium points can be projections of nonsingular phase trajectories of the three-dimensional phase space. The system (85)–(87) has equilibrium states that fill one-dimensional manifolds. Therefore, the problem of equilibrium points falls into two problems: the problem of equilibrium points of the system (85)–(87) in the phase space R1 {v} × R2 {α, ω} and the problem of equilibrium points of the truncated system (86), (87) on the plane R2 {α, ω} or on the cylinder S1 {α mod 2π} × R1 {ω}. 7.1.1. Equilibrium points of third-order system. The desired equilibrium points are given by systems with parameters v1 , v2 , and v3 : π (128) α = + πl, l ∈ Z; , ω = 0; v = v1 , 2 1 π (129) α = − + 2πl, l ∈ Z; ω = − , σ = 0; v = v2 , 2 σ 1 π (130) α = + 2πl, l ∈ Z; ω = , σ = 0; v = v3 . 2 σ Here the parameters v1 , v2 , and v3 are positive numbers. 837
In the phase space R1+ {v} × R2 {α, ω}, systems (128)–(130) define one-dimensional manifolds (lines) entirely filled with equilibrium points of the system (85)–(87) (cf. [246, 260]). 7.1.2. Mechanical interpretation of equilibrium points of three-dimensional phase space. (1) System (128). A body executes a translational motion to any side at constant velocity parallel to a plate. (2) Systems (129), (130). The squared velocity of the center of mass is found from the relation vc2 (v, ω, α) = v2 (1 + σ 2 ω 2 − 2σω sin α). 1 π 1 π 2 v2 , , + 2πl = vc v3 , − , − + 2πl = 0, σ 2 σ 2 the center of mass is immovable. Since dφ vk = , k = 2, 3, dt σ Since
vc2
l ∈ Z,
the body rotates around the immovable center of mass at any constant velocity. 7.1.3. Trivial and nontrivial equilibrium points of second-order system on the plane. The system (86), (87) has equilibrium points on the plane R2 {α, ω}, which are given by the relations α = 2πk, k ∈ Z;
ω = 0,
(131)
α = (2k + 1)π, k ∈ Z; ω = 0, (132) π (133) α = + πl, l ∈ Z; ω = 0, 2 1 π (134) α = − + 2πl, l ∈ Z; ω = − , σ = 0, 2 σ 1 π (135) α = + 2πl, l ∈ Z; ω = , σ = 0. 2 σ Systems (133), (134), and (135), and only they, define the equilibrium points on the phase plane into which manifolds of equilibrium states are orthogonally projected. Hence systems (131) and (132) define points into which are orthogonally projected particular solutions of the form B v(q) = v0 e− m q , v0 = v(0), (136) α(q) ≡ πk, k ∈ Z, ω(q) ≡ 0. The mechanical interpretation of the particular solutions (136) is as follows: the body executes rectilinear translational motion with the angle of attack equal either to 0, or to π. In this case, by the relation d d v = v v, dt dq the velocities of all points of the body decay as t−1 . Because of the simplicity of mechanical interpretation for the stationary motions of the equilibrium points (131)–(135), the latter are called trivial equilibrium states. Along with the trivial equilibrium states, there can exist equilibrium states corresponding to nontrivial particular solutions of the system (85)–(87). Definition. Nontrivial equilibrium states (NES) of the system (86), (87) on the plane R2 {α, ω} are equilibrium points not lying on the lines {(α, ω) ∈ R2 : ω sin α cos α = 0}. 838
7.1.4. Nontrivial equilibrium states of a second-order system corresponding to nontrivial particular solutions. For simplicity, let us consider the system (89), (90). The system (86), (87) can be considered analogously. The necessary condition for the existence of NES is their expression by the systems ω = −a sin α, cos α =
B + −m
B2 m2
+ 4σa(1 + σa)(a2 − n20 ) 2σ(n20 − a2 )
ω = −a sin α, cos α =
B −m −
B2 m2
+ 4σa(1 + σa)(a2 − n20 ) 2σ(n20 − a2 )
and, moreover, either a = a+ = or
n0 −σn0 + 2
(137) ,
(138) ,
σ 2 n20 − 4
n0 (−σn0 − σ 2 n2 − 4). 2 Corollary 7.1. For σn0 < 2, there are no NES. a = a− =
Remark. If the system (86), (87) has NES then they satisfy the equation σ 1 ω 2 + g(α) sin α − ωg(α) = 0, I I
(139)
where
F(α) . cos α In particular, if property (16) holds, then g(α) = AB sin α. g(α) =
Let us consider max
α∈(0,π)
Introduce the notation negative if
g∗ I
g(α) F(α) = max = g∗ . sin α α∈(0,π) sin α cos α
= n2 (under the property (16), n2 = n20 ). The discriminant of Eq. (129) is σn < 2.
(140)
Therefore, under conditions (140), (15), and (13) the system (86), (87) has no NES. As was already mentioned, under condition (16), condition (140) is equivalent to the condition σn0 < 2. Taking (140) into account, for simplicity, let us consider the system (89), (90) under the condition σn0 ≥ 2. Proposition 7.1. (1) If n2 B >− 0, m a+ then the system (89), (90) has NES that are given by system (137) for a = a− . (2) If the inequalities B2 B ≤ 2σ(n20 − a2+ ), ≥ 4σa+ (1 + σa+ )(n20 − a2+ ) m m2 hold, then the system (89), (90) has NES that are given by system (137) for a = a+ . 839
(3) If the inequalities B > 2σ(n20 − a2+ ), m
B2 ≥ 4σa+ (1 + σa+ )(n20 − a2+ ), m2
n2 B >− 0 m a+
hold, then the system (89), (90) has NES that are given by system (137) for a = a+ . (4) If the inequalities B < 2σ(n20 − a2+ ), m
B2 ≥ 4σa+ (1 + σa+ )(n20 − a2+ ), m2
n2 B 2; unstable degenerate nodes if µ2 = 2. (2) The equilibrium points (132) are: saddles if µ1 (2µ2−µ1 ) > 4; repellers if µ2 < µ1 , µ1 (2µ2−µ1 ) < 4; attractors if µ2 > µ1 , µ1 (2µ2 − µ1 ) < 4, and, moreover, under the latter two groups of conditions, these points are foci if µ2 < 2 and nodes if µ2 > 2. (3) The equilibrium points (133) are saddles. (4) The equilibrium points (134) and (135) are always attractors: stable foci if µ1 µ2 < 4; stable nodes if µ1 µ2 > 4; stable degenerate nodes if µ1 µ2 = 4. 7.2.2. Bundles of the phase space and its symmetries. Since we can separate the independent second-order subsystem (86), (87) from the third-order system (85)–(87), the phase trajectories in R1+ {v} × R2 {α, ω} lie on surfaces that are two-dimensional cylinders. In particular, if in the whole phase space, there exists a first integral of the system (86), (87), then it is a function of the variables (α, ω), and, therefore, it defines a family of cylinders in R1+ {v} × R2 {α, ω}. Remark. Since
d d φ = v φ, dt dq 1 2 it follows that in the phase space R+ {v} × R {α, ω} of the system (8)–(10), a first integral depending on all three phase variables corresponds to this first integral. Owing to the reduction described above, it is more convenient to construct the phase portrait of the system (85)–(87) in R1+ {v} × R2 {α, ω} by using the phase portrait of the system (86), (87) in R2 {α, ω}. The latter is not a part of the portrait in R1+ {v} × R2 {α, ω} in the set-theoretic sense, but it is the orthogonal projection of the phase portrait on the plane {v = const}. Therefore, it becomes possible to lift the phase trajectories from the planes to the space R1+ {v} × R2 {α, ω} and to obtain the three-dimensional phase portraits of the desired system. Remark. Since v > 0, the motion is possible only in the domain B = {(α, ω, v) ∈ R3 : v > 0} ⊂ R3 {α, ω, v}. If we formally make the change of variables in the domain B by the formula p¯ = ln v, then the obtained vector field in the phase space R3 {α, ω, p¯} is independent of p¯, i.e., it has a cylindrical nature of a higher order and is orthogonally projected on the whole family of planes {v = const}. In this case, the equilibrium states on the plane R2 {α, ω} completely coincide with the projection union of manifolds of singular points, as well as with projections of nonsingular phase trajectories of the domain B. Remark. To solve the problem of trajectory lift to the three-dimensional phase space completely, it is necessary to reveal the sign of the projection of the vector field in R3 {α, ω, p¯} on the axis p¯ or on the axis v in the domain B. ¯ which is the union of surfaces, where Let us consider the surface M = M0 ∪ M, M0 = {(α, ω, v) ∈ B : Ψ(α, ω) cos−1 α = 0} ¯ is represented (at points where cos α = 0, the surface is complitely defined by continuity), and, in turn, M in the form ¯ = Ms , M s∈Z
where
π + πs}. 2 The listed surfaces are cylinders, which cuts the phase space into domains in each of which the projection of the vector field on the axis p¯ has a fixed sign. On the surface M itself (and only on it), the projection of the vector field on the axis v vanishes. Ms = {(α, ω, v) ∈ B : α =
841
Remark. For any F ∈ Φ and s ∈ Σ, the vector field of the system (86), (87) has the central symmetry property with respect to the points (πk, 0), k ∈ Z, i.e., in the coordinates (α, ω), the vector field changes its direction under the change πk + α πk − α → , k ∈ Z. ω −ω In other words, the vector field of the system (85)–(87) is symmetric with respect to the rays {(α, ω, v) ∈ B : α = πk, ω = 0}. 7.3. Classification of System Phase Portraits on a Two-Dimensional Cylinder for the First Parameter Domain. 7.3.1. An introduction to phase portrait classification. Let us depict the surgery diagram of the vector field of the system (86), (87) near the point (π, 0) (Fig. 44). The behavior of the system field near this point defines many basic topological features of the global phase portrait.
saddle
attractor repeller
region region region
region region region
Fig. 44: Surgery diagram of the system vector field.
We study those dynamical systems which admit the fulfillment of relation (139) only on trivial equilibrium states. Moreover, for simplicity, we assume that inequality (140) holds for any F ∈ Φ and s ∈ Σ. As was already noted, the latter inequality transforms into the inequality σn0 < 2 if F = F0 ∈ Φ and s = s0 ∈ Σ. Since in the general space of physically admissible parameters, M2 = {(µ1 , µ2 ) ∈ R2 : µ1 > 0, µ2 > 0} we mainly study the domain J2 = {(µ1 , µ2 ) ∈ R2 : µ1 > 0, 0 < µ2 < 2} ⊂ M2 . To perform the classification of the system phase portraits on the plane or cylinder, we need to answer the following questions. (1) Existence and uniqueness of trajectories that have infinitely distant points as their α- and ω-limit sets. Lemma 4.13 proves the existence and uniqueness of trajectories going to infinity. Their α- and ω-limit sets are the infinitely distant points (πk, ±∞), k ∈ Z. In this lemma, it is shown that only these points have the property mentioned above. (2) Since the dynamical systems are considered on a cylinder, it is required to reveal the existence of closed trajectories or closed curves of trajectories that are not contractible to a point along the cylinder. Under natural assumptions, Proposition 4.1 (when all conditions of Lemma 4.9 hold) yields clear information on the mutual location of phase curves of the systems (86), (87) and (79), (80). It is easy to show that, in this case, the system (86), (87) has no closed phase characteristics enveloping the phase cylinder. 842
(3) Existence of closed curves of trajectories that are contracted to a point along the cylinder, in particular, the existence of nonisolated periodic trajectories or limit cycles. Note that by the 2π-periodicity of the system vector field in α, the latter problem reduces to finding closed trajectories or closed curves of trajectories only around the equilibrium points of index 1. (4) The main problem of classification of portraits is the problem of behavior of stable and unstable separatrices for existing hyperbolic saddles. These separatrices partition the whole phase plane into domains without equilibrium states. Therefore, the phase portraits are immediately completed. The answer to the latter question requires the proof of a number of auxiliary assertions, which are considered below. Remark. For any F ∈ Φ and s ∈ Σ, let us define the parameter domains as follows: domain I: π
σ F(α) s(α) 2 ≥ for all α ∈ 0, ; (µ1 , µ2 ) ∈ J : m cos α I sin α 2 domain II:
π s(α) σ F(α) 2 (µ1 , µ2 ) ∈ J : there exists α ∈ 0, , for which < , moreover, (π, 0) is a repeller ; 2 m cos α I sin α domain III:
π s(α) σ F(α) 2 (µ1 , µ2 ) ∈ J : there exists α ∈ 0, , for which < , moreover, (π, 0) is an attractor ; 2 m cos α I sin α domain IV: {(µ1 , µ2 ) ∈ M2 : (π, 0) is a saddle}. Note that if F = F0 and s = s0 , then the parameter domains I, II, and III coincide with the domains I, II, and III shown in Fig. 44. 7.3.2. Partition of system phase cylinder into trajectories for first parameter domain. Let us consider the system (86), (87) in the parameter domain I. The following assertion refers to the whole parameter space M2 and is based on the existence of a Poincar´e topographical system in the strip Π. Proposition 7.3. The stable separatrix in the strip Π for the point − π2 , 0 has the origin as its α-limit set. Proposition 7.3 is proved by the PTS method (see Sec. 4). Corollary 7.5. By the central symmetry of the field, the behavior of the analogous separatrices obeys analogous laws. Corollary 7.6. In the strip Π, the system (86), (87) does not have any closed curves composed of trajectories. Proposition 7.4. Let us consider the system (86), (87) in the parameter domain I. Let F(α) sin α ≥ 0 in the strip Π. Then if σ s(α) ≥ gˆ(α), m I where π gˆ (α) = −2F(α), gˆ = 0, 2 it follows that a PTS centered at the point (π, 0) entirely lies in the domain bounded by a nontrivial contact curve (see Proposition 4.2). 843
Corollary 7.7. Let condition (16) hold. Then, obviously, the following inequality holds in the parameter domain I: π π B ≥ σn20 cos α for all α ∈ − , . m 2 2 Moreover, gˆ(α) = AB cos2 α. Therefore, for F = F0 , we are necessarly in the conditions of Proposition 7.4. Proposition 7.5. Let all conditions of Proposition 7.4 hold. Then at least in the parameter domain I, the separatrix entering the point π2 , 0 in the strip Π has the point (π, 0) as its α-limit set. 7.3.3. Topological structure of phase portraits types for first parameter domain. Let us consider the prob lem of the behavior of separatrices emanating from the points − π2 , 0 to the strip Π and from π2 , 0 to the strip Π . The result concerning the behavior of these separatrices is extended to dynamical systems in the part of the parameter domain M2 in which there is no PTS (in particular, in the domain J2 ). For this purpose, let us give the definition of the separatrix behavior index. Definition. The separatrix behavior index (denoted by isp) is the pair k = (k1 , k2 ), where
1 k1 ∈ N0 ∪ 1 + , l ∈ N0 . 4 Moreover, the value of k2 is chosen nonuniquely and can be equal to k − 1 if k1 ∈ N, 1 2 k2 = k1 + 12 if k1 ∈ N0 , k1 − 14 if k1 ∈ N, and
1 if k1 ∈ / N0 . 4 Here and inwhat follows, N0 = N ∪ {0}. By definition, isp = k = (k1 , k2 ) if the separatrix emanating π from − π2 , 0 to the strip Π has the point distant from the point π − 2 , 0 by 2πk1 along the axis α as its ω-limit set, whereas the separatrix emanating from the point 2 , 0 to the strip Π has the point distant from π2 , 0 by 2πk2 along the axis α as its ω-limit set. The latter definition is based on the fact that the separatrices emanating from the points π2 + 2πl, 0 , l ∈ Z lie the domain a part of which boundary is composed of the separatrices emanating from the in π points − 2 + 2πl, 0 for each fixed l ∈ Z. k2 = k1 +
Theorem 7.1. For any isp = k from its domain, the corresponding behavior of the separatrices considered is possible. Before proving Theorem 7.1, we present several auxiliary assertions. Proposition 7.6 characterizes the behavior of the separatrix of the system (79), (80) under the existence of a “perturbation” yielding the system (86), (87). Proposition 7.6. The perturbation of the system (79), (80) the system (86), (87) splits the πyielding , 0 to the strip Π and entering the point separatrix of the system (79), (80) emanating from the point − 2 π 32,0 . Introduce the following notation: B ∗ = max |s(α)|, α∈R
µ∗ = 2
The following relations are obvious for F = F0 and s = s0 : B ∗ = B, 844
µ∗ = µ1 .
B∗ . mn0
Proposition 7.7. For any M > 0 and a sufficiently small value µ2> 0, there exists ε > 0 such that for π ∗ any positive value µ < ε, the separatrix emanating from the point − 2 , 0 to the strip Π extends to the right along the axis α more than by M . Note that the introduction of the parameter µ∗ for each fixed function s does not introduce any restrictions on the function spaces Φ and Σ and the parameter space M2 . Let us begin the proof of Theorem 7.1. Consider the separatrix emanating from the point − π2 , 0 to the strip Π. The phase maps a part of the line Λ−1 onto the line Λ1 . (Here ! flow of the system studied " as previously, Λk = (α, ω) ∈ R2 : α = π2 + πk .) Therefore, let us consider the mapping of the line Λ−1 onto Λ1 . For the comparison system (79), (80) with rotations (see Fig. 4), this mapping is identical at least near the point − π2 , 0 in the domain {(α, ω) ∈ R2 : ω > 0}. Proposition 7.8. For the system (86), (87), the constructed mapping (where it is defined ) is as follows under condition (130): π 3 π, ω2 , ω1 > ω2 . − , ω1 → 2 2 Let us continue the proof of Theorem 7.1. Taking account of the continuous dependence of the mapping constructed in Proposition 7.8 on the parameter µ∗ , let us consider the dependence of the phase portrait type on the parameter µ∗ for sufficiently small µ2 . Assume that the separatrix emanating from the point − π2 , 0 to the strip Π has a limit set distant from the line Λ−1 by 2πk1 , k1 ∈ N0 , along the axis α. Consider the PTS defined by the system (79), (80) going around the point − π2 + 2π(k1 + 1), − σ1 . This system (PTS) is bounded by aπ singular trajectory 1 ∗ to infinity (see Fig. 4). Let us diminish the parameter µ so that the point − 2 + 2πk1 , − σ is no longer the ω-limit of a trajectory, the separatrix, from the separatrix. The π so-called “separation” π set for this 1 1 point − 2 + 2πk1 , − σ to the point − 2 + 2π(k1 + 1), − σ is due to the fact that the angle between the vector fields of the system (86), (87) and the comparison system (79), (80) tends to zero as µ∗ → 0. The case k1 ∈ N0 is considered. Now let us show that in this case, k2 is chosen from the set described above (see Definition of isp). Note that three logical situations are possible for k2 when k1 ∈ N0 . Recall that on the phase cylinder, the systems (79), (80) and (86), (87) has infinitely distant limit points (−0, −∞), (+0, +∞), (π + 0, −∞), and (π − 0, +∞); moreover, the first two of them are attracting, whereas the last two are repelling. Proposition 7.9. The separatrix studied can have the infinitely distant point − π2 + 2 l + 14 π − 0, −∞ as its ω-limit set. Proof. Let us project the phase plane on the Riemann or Poncar´e sphere [39, 264]. By the smooth dependence of the separatrix studied on the parameter µ∗ , for the critical value of the parameter under which there is a surgery of the rough ω-limit set of the separatrix, the infinitely distant point (on the plane) or the north pole (on the Riemann or Poncar´e sphere) becomes a new ω-limit set. Indeed, if the north pole is not the ω-limit set of this separatrix under the critical value of the parameter, then this separatrix has another rough (examined above) ω-limit set. Since under a small deformation, the latter ω-limit set is preserved, we have a contradiction to the statement that the critical value of the parameter µ∗ corresponds to the rough ω-limit set. The proposition is proved. For a more general assertion, including that on strictly monotone fields, see Sec. 4 and [264]. Let us complete 7.1. Now let k1 = l + 14 , l ∈ N0 . Then the separatrix emanating π the proof of Theorem from the point 2 , 0 to the strip Π has no other ω-limit sets except for the point 1 1 π , k2 = k 1 + , k1 ∈ / N0 . − + 2πk2 , − 2 σ 4 Note π that these ω-limit sets are rough because of the hyperbolicity of the asymptotically stable points − 2 + 2πl, − σ1 . 845
Now to prove Theorem 7.1, it suffices to deform a little the parameters µ∗ and µ2 of the dynamical system (86), (87) considered. Indeed, if k2 is fixed, then k1 can be equal either to k2 ± 12 (the rough ω-limit set moves) or (by Proposition 7.9) to k2 + 14 (the nonrough ω-limit set). Theorem 7.1 is proved. 7.3.4. A complete classification of the phase portraits of dynamical system for first parameter domain on two-dimensional cylinder. According to Theorem 7.1, we can perform a complete classification of the phase portraits of the system (86), (87) when its parameters vary in the domain I. As is shown above, there are infinitely many such portraits. As a consequence, let us consider the comparison system (79), (80) for the system (86), (87). It describes a conservative system, the physical pendulum on the plane. Then among the systems of the form (86), (87) that differ from (79), (80) sufficiently little, there exist systems with any of the phase portraits described above. In other words, any sufficiently small perturbation yielding a system of the form (86), (87) makes a surgery of the global topological type of the physical pendulum infinitely many times. Some of the phase portraits (the index isp assumes first ten values (0, 1/2), (1/4, 1/2), (1, 1/2), (1, 3/4), (1, 3/2), (5/4, 3/2), (2, 3/2), (2, 7/4), (2, 5/2), and (9/4, 5/2)) are shown in Figs. 7–16, respectively. In this case, the functions F and s satisfy conditions (15), (13), and (140). The further possible values of the index isp are as follows: (3, 5/2), (3, 11/4), (3, 7/2), etc. In this way, we perform the complete classification of the phase portraits of the system (86), (87) for the parameter domain I. Note that many assertions of this section also hold in wider parameter domains. In conclusion, we want to mention the general property of the body motion. Let us consider the parameter domain I. A dimensionless moment small as compared with the dimensionless force corresponds to it. In this domain, for almost any initial condition in the sense of measure theory, in a finite sufficiently large time, the rigid body tends to an exponentially stable stationary motion of the following form: the body moves around the immovable center of mass at a constant angular velocity. In the material just presented, we begin to consider the model variant of the problem of the free plane-parallel drag of a body in a medium. We present an auxiliary qualitative analysis of the systems of differential equations describing such a body motion in a certain domain of nonzero measure in the parameter space. On its basis, we obtain a new two-parameter family of phase portraits consisting of an infinite set of different portrait types. In this case, there are no auto-oscillations, and for almost any initial conditions, all trajectories tend to asymptotically stable equilibrium states. The next material continues the global qualitative analysis of the dynamical system in the quasi-velocity space. In this case, we study another domain of nonzero measure in the system parameter space and obtain a new two-parameter family of phase portraits with limit cycles. Therefore, under certain conditions, there can arise auto-oscillations in the system considered. 7.4. A Classification of Portraits for Second and Third Parameter Domains. 7.4.1. Problem of separatrix behavior. The previous material was presented in sufficient detail. In what follows, we omit proofs of many assertions. They follow from the qualitative material prepared in the previous sections. Let us consider the system (86), (87) in the parameter domains II and III (in this case, we formally replace α by −α). As previously, the key problem of the portrait classification is the problem of stable and unstable separatrix behavior of the existing hyperbolic saddles. Recall that limit cycles for the system studied can exist only in the strip Π (Sec. 4). Let us consider the key problems of the global behavior of the following separatrices: (a) the separatrix emanating from the point − π2 , 0 to the strip Π; (b) the separatrix entering the point − 3π 2 , 0 from the strip Π(− 3 π,− π ) ; 2
846
2
(c) the separatrix emanating from the point − π2 , 0 to the strip Π(− 3 π,− π ) . 2
2
By independent behavior of separatrices we mean that the separatrices considered have limit sets independently chosen from the domain of definition of possible limit sets of separatrices, all logically possible limit sets of separatrices arising as “candidates” for true limit sets after taking account of the possible location of the system isoclines and the character of existing singular points. Lemma 7.1. The global behavior of any two separatrices from items (a)–(c) is independent. The behavior of the third separatrix is determined by the behavior of the other two. As a pair of key separatrices whose behavior is independent, let us choose the separatrices (a) and (b). Definition. The index k1 of the separatrix (a) is the rational number chosen from the set
1 r ∈ Q : r = m, r = + m, m ∈ N0 . 4 π 1 if r ∈ N0 and the point = r if the separatrix (a) has the point − + 2πr, We say that k 1 2 σ π / N0 as its ω-limit set. − 2 + 2πr − 0, +∞ if r ∈ Definition. The index k2 of the separatrix (b) is the natural number j chosen from the set {j ∈ N : j = 1, 2, 3, 4, 5}.
We say that k2 = j if the separatrix (b) has the point (−π, 0) or the cycle if j = 1; the point − π2 , 0 if j = 2; the point (0, 0) if j = 3; the point (−π − 0, −∞) if j = 4; the point (−2π, 0) if j = 5 as its α-limit set. Therefore, we see that the behavior of the separatrix (c) depends on the indices k1 and k2 , i.e., on the concrete behavior of the separatrices (a) and (b). Remark. If we fix the index k1 , then, in particular, the index k2 can be chosen from a narrower set described in the previous definition. The direct product of the sets described in the previous two definitions is maximal in the sense of possible definition of the pair of indices k1 and k2 . 7.4.2. Main theorem on classification of phase portraits. Denote by k2∗ the index k2 in the case where the latter is chosen from the truncated domain of definition in connection with the previous remark and compose the direct product of the indices k1 and k2∗ . For this purpose, let us consider the index k = (k1 , k2∗ ). Theorem 7.2. For any k from its domain, the corresponding behavior of the separatrices (a) and (b) is possible. Therefore, we construct a two-parameter family of phase portraits containing portraits with limit cycles and also an infinite set of phase portraits with different qualitative properties. The theorem allows us to make the following conclusion: any sufficiently small perturbation yielding the desired system in the parameter domains II and III for the system describing the physical pendulum on the plane makes a surgery of the global type for the Hamiltonian phase portrait of the physical pendulum infinitely many times. Some of the portraits (the index k assumes the values (0, 1)∗ , (0, 3)∗ , (0, 4)∗ , (0, 2)∗ , (0, 1), (0, 3), (0, 2), and (1/4, 3)∗ ) are shown in Figs. 17–24. In contrast to the just constructed family, the two-parameter family constructed in [249,261,267,279] contains no limit cycles. But these families are united by the fact that the pair of independent indices (in this case, k1 and k2∗ ) “coding” the topological type of the phase portrait corresponds to each independent pair of dimensionless parameters µ1 and µ2 . In the latter case, we need to stipulate the phase portraits with limit cycles existing in the family. Such portraits can have topological types different from the portraits with the same indices k, but without limit cycles. In the latter case, we can somehow label the indices k and speak of the labelled phase portraits (for example, in the case of a cycle, we have the index (k1 , k2 )∗ : the phase portrait depicted in Fig. 17 has the labelled index (0, 1)∗ ). 847
7.5. Structure of System Phase Portrait in the Fourth Parameter Domain. 7.5.1. Structure of phase portrait with “richest” equilibrium states. Depending on the fulfilment of one or another property from Proposition 7.1, the system (86), (87) has “more” or “less” additional singular points on the phase cylinder corresponding to NES’. In each concrete case, the existence of such points and their stability are studied separately. For the parameter subdomain of the domain IV, when there exist NES’ given by both systems ((137) and (138)), the phase portrait is shown in Fig. 45.
Fig. 45: System portrait in the case where there exist sufficiently many equilibrium states.
Proposition 7.10. The system (86), (87) considered in the parameter domain IV has NES’ lying only in the strip Π that correspond to repelling or attracting and also saddle singular points. Therefore, Proposition 7.10 concerns the case where the studied system has the maximally possible number of additional singular points. In any case, NES’ correspond to particular solutions of the system (8)–(10), (104) that are expressed through the formulas of Corollary 7.3. Therefore, the latter particular solutions of the system (8)–(10), (104) can be made orbitally stable, as well as orbitally unstable. Continuing the qualitative analysis, we note that if the system (86), (87) is considered in the parameter domain IV, then it has no periodic trajectories contractible to a point along the phase cylinder. Let us study the problem of the behavior of stable and unstable separatrices that the existing hyperbolic saddles have. Note that in the domain W = {(α, ω) ∈ Π : ω > 0}, there exit the following three NES’ of the system (86), (87) (α → −α): one repelling equilibrium state, one attracting equilibrium state, and one saddle equilibrium state. Proposition 7.11. set. (1) In the strip Π, for the point − π2 , 0, the stable separatrix has the origin as its α-limit (2) In the strip Π, for the point − π2 , 0 , the unstable separatrix has the point − π2 , σ1 as its ω-limit set. (3) In the strip Π , for the point − π2 , 0 , the stable separatrix has the repelling NES in the domain W as its α-limit set. 848
(4) In the strip Π , for the point − π2 , 0 , the unstable separatrix has the attracting NES in the domain R2 {α, ω} \ W as its ω-limit set. (5) In the strip Π(−π,−π/2) , for the point (−π, 0), the stable separatrix has the repelling NES in the domain W as its α-limit set. (6) In the strip Π(−π,−π/2) , for the point (−π, 0), the unstable separatrix has the attracting NES in the domain W as its ω-limit set. (7) In the domain W , for the saddle NES, two stable separatrices have the origin and the repelling NES in the domain W as their α-limit sets. (8) In the domain W , for the saddle NES, two unstable separatrices have the point − π2 , σ1 and the attracting NES in the domain W as their ω-limit sets. The problem of limit sets for other key trajectories is obviously solved owing to Proposition 7.11 (see Fig. 45). 7.5.2. Classes of rigid body motion trajectories on plane. In accordance with the global phase portrait (see Fig. 45), the typical trajectories of motion of the rigid body point D on the plane are divided into the following classes. The unstable stationary rigid body motion with zero angle of attack (the center of mass is behind) and also the unstable stationary motion corresponding to the repelling NES have the following limit motions: (1) the body asymptotically tends to the rotation around the immovable center of mass at constant angular velocity; (2) the body asymptotically tends to the stationary motion corresponding to the attracting NES. In conclusion, we note that the considered classes of motions (since the dynamical system is absolutely structurally stable in the quasi-velocity space) have the important (absolute) roughness property. Moreover, the domain of system physical parameters is of a sufficiently “large” nonzero measure. 8. Methods for Analyzing Zero Mean Variable Dissipation Dynamical Systems in Spatial Dynamics of a Rigid Body In this section, we consider the possibilities of extending the results of the plane dynamics of a rigid body interacting with a medium to the spatial case. We analyze the problems of a spherical pendulum placed in the over-running medium flow and a spatial body motion under the existence of a certain nonintegrable constraint and also show the mechanical and topological analogies of the latter two problems. 8.1. Statement of Problem of Spatial Body Motion in a Resisting Medium under Streamline Flow Around. The conjectures presented in [184, 185] (see also [60, 128, 132, 440]), which concern the medium properties, are reflected in constructing the spatial dynamical model of interaction of a medium with a body. In this connection, there arises the possibility of formalizing the model assumptions and obtaining the complete system of equations. (1) The whole interaction of the medium with the body is concentrated on the part of the body surface, which has the form of a convex plane domain P (Fig. 46). (2) Since the interaction obeys the streamline flow-around laws, the force S of this interaction is directed along the normal to the domain, and, moreover, the point N of application of this force is determined by only one parameter, the angle of attack α, which is made by the velocity vector v of the plate point D and the exterior normal at this point (the line CD). The point D is the intersection point of the middle perpendicular dropped from the center of mass C to the plane P . Therefore, DN = R(α). (3) Assume that the value of the resistance force S has the form S = s1 v2 , where v is the modulus of the velocity vector of the point D, and the resistance coefficient s1 (as in the plane case; see (13)) is a function of only the angle of attack α: s1 = s1 (α). Therefore, as previously, we consider such an “extension” of the problem, which was mentioned in Sec. 1. 849
Fig. 46: Spatial model of the dynamical action of the medium on the body.
(4) As in the plane case, along the line CD, an additional force T can act on the body; as before, it is called the “traction force.” The introduction of this force is used for ensuring some given classes of motion (in this case, T is the reaction of possible imposed constraints). In the case where there is no external force T, the body executes a spatial free drag in the resisting medium (see Sec. 9). (5) Denote by Dxyz the coordinate system related to the body. This coordinate system related to the point D is chosen such that the tensor of inertia in this system has the diagonal form: diag{I1 , I2 , I3 }. Assume that the mass distribution is such that the longitudinal principal axis of inertia coincides with the axis CD (this is the axis Dx), whereas the axes Dy and Dz lie in the plane P and compose the right coordinate system. Moreover, we consider the case of the dynamically symmetric rigid body, i.e., I2 = I3 . (6) In this case, to describe the body position in the space, we choose the Cartesian coordinates (x0 , y0 , z0 ) of the point D and three angles (θ, ψ, φ), which are defined similar to the navigational angles as follows. Let us represent the turn from the system Dx0 y0 z0 to the system Dxyz as a composition of three turns T3 (φ)T2 (ψ)T1 (θ) under which, first, the frame (ex0 , ey0 , ez0 ) is turned around the vector ex0 by the angle θ (T1 (θ) is executed): T1 (θ) (1) (ex0 , ey0 , ez0 ) −−−→ ex0 , e(1) y 0 , e z0 , (1) (1) (1) then the frame ex0 , ey0 , ez0 is turned around the vector ey0 by the angle ψ (T2 (ψ) is executed): (2) (1) (1) T2 (ψ) ex0 , e(1) y0 , ez0 −−−→ ex0 , ey0 , ez , (2) (1) and finally, the frame ex0 , ey0 , ez is turned around the vector ez by the angle φ (T3 (φ) is executed):
T3 (φ) (1) e(2) x0 , ey0 , ez −−−→ (ex , ey , ez ).
In this case, the vectors having the components in the frame (ex , ey , ez ) obtain new coordinates in the basis (ex0 , ey0 , ez0 ). In the basis (ex , ey , εz ), such a transformation is given by the matrix cos ψ cos φ − cos ψ sin φ sin ψ cos θ sin φ + sin θ sin ψ cos φ cos θ cos φ − sin θ sin ψ sin φ − sin θ cos ψ ; sin θ sin φ − cos θ sin ψ cos φ sin θ cos φ + cos θ sin ψ sin φ cos θ cos ψ then the phase state of the system is characterized by twelve quantities (x0 , y0 , z0 , θ; ψ; φ; x0 , y0 , z0 , θ, ψ, φ). 850
Let us consider the spherical coordinates of the velocity vector endpoint of the point D(v, α, β) in which the angle β is measured from the axis Dy in the plane P up to the line (DN ) which is the intersection of two planes one of which contains the vector v and the axis Dx and the other is the plane P . The latter spherical coordinates and also the components of the angular velocity are expressed through the phase variables (x0 , y0 , z0 , θ; ψ; φ; θ, ψ, φ) via nonintegrable relations. Therefore, the phase state of the system is determined by the functions (v, α, β, Ωx , Ωy , Ωz , x0 , y0 , z0 , θ, ψ, φ), and the first six functions are considered as quasi-velocities of the system. Here the tuple (Ωx , Ωy , Ωz ) is defined as Ω = Ωx ex + Ωy ey + Ωz ez , where Ω is the absolute angular velocity vector of the rigid body. Since the generalized forces are independent of the body position in the space, the coordinates (x0 , y0 , z0 , θ, ψ, φ) are cyclic. This allows us to consider the dynamical part of the equations of motion as an independent subsystem. Analogously to the plane case, let us introduce the sign-alternating auxiliary function s(α) = s1 (α) sign cos α. By the theorem on the motion of the center of mass in the space in projections to the related axes (x, y, z) and the theorem on the change of the kinematic moment with respect to these axes, we obtain the following complete system of differential equations in the dynamical quasi-velocity space R1+ {v} × T2 {α, β} × R3 {Ωx , Ωy , Ωz }: s(α) 2 T − v , v˙ cos α − αv ˙ sin α + Ωy v sin α sin β − Ωz v sin α cos β + σ(Ω2y + Ω2z ) = m m ˙ sin α sin β + Ωz v cos α − Ωx v sin α sin β − σΩx Ωy − σ Ω˙z = 0, v˙ sin α cos β + αv ˙ cos α cos β − βv ˙ sin α cos β + Ωx v sin α cos β − Ωy v cos α − σΩx Ωz − σ Ω˙y = 0, v˙ sin α sin β + αv ˙ cos α sin β + βv (141) I1 Ω˙x + (I3 − I2 )Ωy Ωz = 0, I2 Ω˙y + (I1 − I3 )Ωx Ωz = −zN s(α)v2 , I3 Ω˙z + (I2 − I1 )Ωx Ωy = yN s(α)v2 .
The coordinates of the point N in the system (ex , ey , ez ) take the form 0, yN (α, β), zN (α, β) , where yN (α, β) = R(α) cos β, zN (α, β) = R(α) sin β. Let us complement system (141) by the kinematic relations θ˙ =
1 [Ωx cos φ − Ωy sin φ], cos ψ
ψ˙ = Ωx sin φ + Ωy cos φ, sin ψ [Ωy sin φ − Ωx cos φ], φ˙ = Ωz + cos ψ x˙0 = cos ψ cos φ cos α − cos ψ sin φ sin α cos β + sin ψ sin α sin β, v y˙0 = (cos θ sin φ + sin θ sin ψ cos φ) cos α + (cos θ cos φ − sin θ sin ψ sin φ) sin α cos β v + sin θ cos ψ sin α sin β, z˙0 = (sin θ sin φ − cos θ sin ψ cos φ) cos α + (sin θ cos φ + cos θ sin ψ sin φ) sin α cos β v + cos θ cos ψ sin α sin β.
(142)
As before, we use the following notation: σ is the distance DC and m is the mass of the body. 851
The dynamical system (141) contains the functions R(α) and s(α); to describe them qualitatively, as before, we use the existing experimental information about the properties of the streamline flow around (see Sec. 1). The function R(α) satisfies condition (11), whereas the function s(α) satisfies condition (13). 8.2. Case of Body Motion in a Medium under a Certain Nonintegrable Constraint and Beginning of Qualitative Analysis. As in the plane case, we consider the class of motions under which identity (69), being a nonholonomic constraint, holds. 8.2.1. On analytical integral. In the case of the dynamically symmetric rigid body, system (141) has the analytical integral (143) Ωx ≡ Ωx0 = const, i.e., the generalized forces admit a body self-rotation around the longitudinal dynamical symmetry axis. 8.2.2. On appearance of an independent subsystem. Let us use (see Sec. 6) the methodological tool of reducing the system order, i.e., from the first equation of (141), let us express the function T so that the total derivative of v by system (141) vanishes. As a result of this order reduction, system (141) has an independent subsystem of the form ˙ sin α sin β + Ωz v cos α − Ωx0 v sin α sin β − σΩx0 Ωy − σ Ω˙z = 0, αv ˙ cos α cos β − βv ˙ sin α cos β + Ωx0 v sin α cos β − Ωy v cos α + σΩx0 Ωz + σ Ω˙y = 0, αv ˙ cos α sin β + βv I2 Ω˙y + (I1 − I2 )Ωx0 Ωz = −zN s(α)v2 ,
(144)
I2 Ω˙z + (I2 − I1 )Ωx0 Ωy = yN s(α)v2 in which the parameter v is added to constant parameters. 8.2.3. Case of zero projection of angular velocity on longitudinal axis and case of analytic system. Let us consider the trajectories of motion of system (144) on the level of integral (143) for Ωx0 = 0. In this case, it takes the form ˙ sin α sin β + Ωz v cos α − σ Ω˙z = 0, αv ˙ cos α cos β − βv ˙ sin α cos β − Ωy v cos α + σ Ω˙y = 0, αv ˙ cos α sin β + βv (145) I2 Ω˙y = −F(α) sin βv2 , I2 Ω˙y = F cos βv2 . The first two equations reduce to the form σ vF(α) = 0, I β˙ sin α − Ωy cos α cos β − Ωz cos α sin β = 0. α˙ cos α + Ωz cos α cos β − Ωy cos α sin β −
(146)
If conditions (12) and (14) (or (16)) hold, then, as previously, introducing the notation n20 , instead of (146), we have the following analytic system (n20 = AB/I2 ): α˙ + Ωz cos β − Ωy sin β − σn20 v sin α = 0, β˙ sin α − Ωy cos α cos β − Ωz cos α sin β = 0;
(147)
let us complement it by the equations Ω˙y = −n20 v2 sin α cos α sin β, Ω˙z = n20 v2 sin α cos α cos β.
(148)
The system (147), (148) is closed. Note that system (147) is equivalent to (146) only outside the manifold (149) O = {(α, β, Ωy , Ωz ) ∈ R4 : cos α = 0}. 852
Let us project the angular velocity to the movable coordinate system Dz1 z2 (turning the system Dyz by the angle −β) such that z1 = Ωy cos β + Ωz sin β,
z2 = Ωz cos β − Ωy sin β.
(150)
˙ 2 , system (146) is equivalent to the following system outside the manifold O : In this case, since z˙1 = βz v F(α) , I2 cos α cos α F(α) 2 z˙2 = , v − z21 I2 sin α cos α , z˙1 = z1 z2 sin α cos α , β˙ = z1 sin α
α˙ = −z2 + σ
(151)
(152)
where O = {(α, β, Ωy , Ωz ) ∈ R4 : sin α cos α = 0}.
(153)
Under condition (16), the system (151), (152) takes the form of the analytic system α˙ = −z2 + σn20 v sin α, z˙2 = n20 v2 sin α cos α − z21
cos α , sin α
(154)
cos α , sin α (155) cos α . β˙ = z1 sin α Outside the manifold O , from the dynamical system (145), the independent third-order subsystem (151) is separated; this system is considered not only on the three-dimensional cylinder S1 {α mod 2π} × R2 {z1 , z2 }, but on the tangle bundle to the two-dimensional sphere (see Sec. 2). Since we have the “separation” of the third-order system, the phase space of our system already has a number of bundles [405]. z˙1 = z1 z2
8.2.4. System equilibrium points and stationary motions corresponding to them. The set of equilibrium points of system (146) to which the stationary motions correspond is partitioned into two parts. First, there exist the singular points {α = 0 mod 2π, Ωy = Ωz = 0},
(156)
corresponding to those stationary motions under which the body moves translationally along the line CD to one side or another. Second, formally, there exists the two-dimensional manifold O of singular points to which stationary motions discussed in Sec. 6 correspond. 8.2.5. On violation of uniqueness theorem. On the manifold O , it is impossible to resolve system (145) ˙ Formally, the uniqueness theorem is violated on O . uniquely with respect to α, ˙ β. As was mentioned, system (145) is equivalent to (141) only outside O . But, in fact, the uniqueness theorem is violated only on the manifold O ⊂ O , since the vector field of (151) is not defined on O \ O because of the degeneration of the spherical coordinates (v, α, β) of the point D. On O (where we define system (151) by continuity), we have the violation of the uniqueness theorem in the following sense: a nonsingular phase trajectory of system (145) passes through almost any point of O and intersects O at a right angle; also, there exists a phase trajectory coinciding with the above point at all instants of time. But physically, these are different trajectories, since different values of the traction T (reaction of the imposed constraint) correspond to them. Let us show this. 853
To support a constraint of the form (69), it is necessary to choose the reaction T in the form mσ 2 2 2 F(α) tan α . (157) Tv (α, Ωy , Ωz ) = σm(Ωy + Ωz ) + v s(α) − I2 If
F(α) = L, α→ 2 cos α then for α = π2 , the needed value of traction is found from the relation π mσ 2 , Ωy , Ωz = σm(Ω2y + Ω2z ) − Lv , Tv 2 I2 where the quantities Ωy and Ωz are arbitrary. On the other hand, supporting the rotation around a certain point W by using the traction T, we need to choose the traction in the form mv2 , (158) Tv ≡ R0 where R0 is the distance CW . In general, relations (157) and (158) define different values of T, which proves the remark. If |L| = +∞, then the instantaneous phase velocity near the manifold O is infinite. limπ
8.2.6. Symmetries of system vector field in quasi-velocity phase space. The vector field of system (151) has the following three types of symmetries. (1) The central symmetry. Near (πk, 0, 0), k ∈ Z, in the space R2 {α, z2 , z1 }, such a symmetry arises since the vector field in the coordinates (α, z2 , z1 ) alternates its sign under the change πk + α πk − α −z2 → z2 . −z1 z1 (2) A certain mirror symmetry. With respect to the planes Λi , i ∈ Z, where π Λi = (α, z2 , z1 ) ∈ R3 : α = + πi , 2 such a symmetry arises since the α-component of the system vector field in the coordinates (α, z2 , z1 ) is preserved under the change π/2 + πi − α π/2 + πi + α → , z2 z2 z1 z2 whereas the z2 - and z1 -components alternate their signs. (3) The symmetry with respect to the plane {(α, z2 , z1 ) ∈ R3 : z1 = 0}; precisely, the z2 - and α-components of the system vector field are preserved under the change α α z2 → z2 , z1 −z1 whereas the z1 -component alternates its sign. 8.3. On Transcendental Integrability of the System. This subsection is devoted to studying the possibilities of complete integration of the dynamical system considered. Here we present first integrals of system (154) expressed through elementary functions and also discuss the way of integrating the general system (141) [405, 416]. 854
8.3.1. Theorem on complete integrability. Theorem 8.1. System (154) has a complete tuple of transcendental first integrals. The system (154), (155) is also completely integrable, and two its first integrals are first integrals of system (154) [346]. One of the first integrals of system (154) has the form z22 + z21 − σn20 v sin α = C1 . z1 sin α
(159)
8.3.2. Remarks on integrability of conservative system. Denote by (154) system (154) for σ = 0. Proposition 8.1. System (154) is a conservative system having the following two first integrals: z1 sin α = C10 , z21 + z22 + n20 v2 sin2 α = C20 ,
C10 , C20 = const.
(160)
Despite the simple character of Proposition 8.1, it is necessary to make the following important remark. Remark. Obviously, the ratio of first integrals (160) is the first integral of (154) . But none of the functions (160) (if we formally subtract σn20 vz2 sin α vanishing for σ = 0 from the second function) is a first integral of (154). However, the ratio of the latter two functions is a first integral of system (154) for any σ. 8.3.3. Search for additional system first integrals. Since system (154) has a variable dissipation and is analytic, we succeed in searching for two other additional integrals in explicit form for it. The following identity holds: 1 (161) u1 = {C1 ± G}. 2 Here G = C12 − 4[u22 − σn20 vu2 + n20 v2 ], u1 = z1 τ, u2 = z2 τ, τ = sin α (to seek additional integrals, we use the transcendental first integral (159)). The quadrature for searching for the desired integral connecting the quantities u2 and τ has the form (σn20 v − u2 )du2 dτ . (162) = τ 2[u22 − σn20 vu2 + n20 v2 ] − C21 (C1 ± G) If w1 = u2 −
σn20 v , 2
then the right-hand side of (162) takes the form σn2 v ( 20 − w1 )dw1 2[w12 − n20 v2
σ 2 n20 −4 ] 4
−
Quantity (163) falls into two parts: σn20 v 2
− (1)
where
=
(1)
here,
dw1 , G1
± G)
.
(163)
,
(2)
=
(2)
C1 2 (C1
1 2 2 dw1
G1
;
2 2 C1 2 2 2 σ n0 − 4 − (C1 ± G). G1 = 2 w1 − n0 v 4 2 855
If a = n20 v2 then
σ 2 n20 − 4 ¯2 = C12 − 4(¯ x − a), , x ¯ = w12 , y 4 1 y + C1 | + const. = ln |¯ 2 (2)
Furthermore,
=± (2)
C12
For definiteness, let =± (1)
if σn0 < 2;
d¯ y . (¯ y + C1 ) C12 − y ¯2 + 4a
+ 4a ≥ 0. Then
1 C1 y ¯ + C12 + n20 v2 (σ 2 n20 − 4) + const arcsin n20 v2 4 − σ 2 n20 (¯ y + C1 ) C12 + n20 v2 (σ 2 n20 − 4) (1)
1 =∓ C1 (¯ y + C1 )
C12 − y ¯2 + const
if σn0 = 2; ∓ (1)
n v σ 2 n2 − 4 + G 1 C 0 1 0 1 ln =− 2 + y ¯ + C1 2n0 v2 σ 2 n20 − 4 n0 v σ 2 n20 − 4 n v σ 2 n2 − 4 − G C 1 0 1 0 1 + ln + 2 + const 2 2 2 2 2 y ¯ + C1 2n0 v σ n0 − 4 n0 v σ n0 − 4
if σn0 > 2. 8.3.4. Search for second additional integral. The system of additional integrals found above and being a transcendental function of the phase variables, together with (159), composes a complete tuple of first integrals of the system (154). For the system (154), (155), one more first integral is necessary. Remark. Everywhere above (see the previous subsection), for C1 it is necessary to substitute the left-hand side of relation (159). To seach for the last integral of the system (154), (155), we note that since dz1 = z2 , dβ it follows that du1 + [−u2 + σn20 v] = u2 . dβ Therefore,
du1 = ± σ 2 n40 v2 − 4[u21 − C1 u1 + n20 v2 ], dβ and hence the desired quadrature takes the form du1 = β + C3 , C3 = const. ∓ 4 2 2 σ n0 v − 4[u21 − C1 u1 + n20 v2 ] 856
(164)
(165)
The left-hand side of (165) (without sign) has the form σn2 v
(u2 − 20 )2 1 . arcsin 2 2 C1 + n20 v2 (σ 2 n20 − 4)
(166)
After substitution, we have the desired invariant relation: cos2 [2(β + C3 )] =
1 , G2
(167)
where G2 = [u22 − σn20 v]2 + 2[u22 − σn20 vu2 ] [u21 + n20 v2 ] + [u21 − n20 v2 ]2 + σ 2 n40 v2 u21 . Example. If σn0 = 2, then relation (167) takes the form cos2 [2(β + C3 )] = As an odd function of ζ=
(z2 − n0 v sin α)z1 . (z2 − n0 v sin α)2 + z21
(168)
z2 − n0 v sin α , z1
for ζ = 1, the right-hand side has the global maximum equal to 1/2. 8.3.5. On pendulum with variable dissipation. The system (151), (152) is also a pendulum system with zero mean variable dissipation. The motion is executed under the action of the following two forces: the potential force F(α) I2 and the linear-in-velocity force d F(α) v σ α I2 dα cos α with variable coefficient. In a certain subspace, this coefficient has a strictly positive sign, and, therefore, energy pumping from the medium occurs here. In the other subspace, this coefficient has a strictly negative sign. Therefore the force scatters the energy, forcing the body to damp its motion. The latter remarks show that (as in the plane case) we are dealing with a dissipative zero mean variable dissipation system. The phase space is decomposed into a union of alternating domains in each of which there is a dissipation of only one sign. Remark. In the case of the system (151), (152), the search for integrals reduces to integrating the Riccati equations whose solutions are not expressed in elementary functions in the most general case. 8.4. Problem on Spatial Pendulum in an Over-Running Medium Flow. Analogously to the plane case, let us consider the problem of a spatial pendulum placed in an over-running medium flow. Let a convex plane domain be clamped perpendicularly to the holder by a spherical hinge, and let it be in an over-running medium flow, which moves at a constant velocity v∞ = 0. Assume that the holder does not create any resistance (Fig. 47). The total force S of the medium flow action on the body is directed parallel to the holder, and the point N of application of this force is determined by only one parameter, the angle of attack α measured between the velocity vector vD of the point D with respect to the flow and the holder. Therefore the force S is directed along the normal to the side opposite to the direction of the velocity vD and passes through a certain point N of the plane domain displaced from the point D forward with respect to the direction of vD . Such conditions arise in using the model of streamline flow around spatial bodies. 857
Fig. 47: Spherical pendulum in a homogeneous over-running medium flow.
The vector e determines the orientation of the holder; then 2 e, S = s1 (α)vD
where the resistance coefficient has the form s1 = s1 (α) = s(α) sign α cos α. Let Ox0 y0 z0 be the immovable coordinate system. The direction of the over-running flow coincides with the direction of the axis x0 . Let us relate the coordinate system Dxyz to the body, where the axis x is directed along the holder and the axes y and z are rigidly related to the plane domain. The coordinates of the point N in the system Dxyz have the form (0, yN , zN ). Analogously to the problem of the free body motion, we introduce the function R(α) and also the angle β measured in the plane Dyz. In this case, for simplicity, let property (16) hold. For any admissible function R(α), the analysis is performed analogously. If the body is dynamically symmetric (I1 and I2 = I3 are principal moments of inertia in the system Dxyz) and Ωx , Ωy , and Ωz are projections of the angular velocity in the system Dxyz, then the equations of motion take the following form, which is analogous to (148): 2 sin α cos α sin β, Ωy = −n20 vD
(169)
2 Ωz = n20 vD sin α cos α cos β.
The resistance force admits the existence of the first integral (143), and, in this case, the condition Ωx0 = 0 is taken into account in Eqs. (169) (for simplicity, we now consider only this case). Let us introduce the angles (θ, ψ) determining the orientation of the pendulum. The angle θ is measured from the axis x0 to the holder, whereas ψ is measured from the projection of the holder on the plane Oy0 z0 to the axis y0 . Then cos θ = cos ψ cos φ, 858
sin θ cos ψ = cos ψ sin φ,
sin θ sin ψ = sin ψ.
(170)
8.4.1. Complete system of equations. The relations connecting (vD , α, β) and (θ, ψ, Ωy , Ωz ) (l is the holder length) have the form vD cos α = −v∞ cos θ, vD sin α cos β = lΩz + v∞ sin θ cos ψ, (171) vD sin α sin β = −lΩy − v∞ sin θ sin ψ. By the first three relations, from (142), we have sin φ , θ˙ = −Ωy cos ψ sin φ , φ˙ = Ωz + Ωy sin φ cos ψ ψ˙ = Ωy cos φ,
(172)
whence we easily deduce that ψ˙ , cos φ sin φ sin ψ . Ωz = φ˙ − ψ˙ cos φ cos ψ Using properties (170) and (173), we have the identities Ωy =
(173)
sin θ cos ψ, cos θ (174) sin θ ˙ ˙ sin ψ. Ωz = θ cos ψ − ψ cos θ Equations from (169), (171), and (174) compose a complete system for determining the pendulum motion on the level of the integral Ωx0 = 0. Ωy = θ˙ sin ψ + ψ˙
8.4.2. System of differential equations and topological analogy. Starting from three groups of equations two of which are differential and the third is algebraic, it easy to prove the following proposition. Proposition 8.2. The complete system of equations of the pendulum motion on the tangent bundle of the two-dimensional sphere has the following form (cf. the corresponding equations from Sec. 2): sin θ 2 = 0, sin θ cos θ − ψ˙ 2 θ¨ + ln20 v∞ θ˙ cos θ + n20 v∞ cos θ 1 + cos2 θ ψ¨ + θ˙ψ˙ + ln20 v∞ ψ˙ cos θ = 0. cos θ sin θ
(175)
As in the case of a free body, system (175) has symmetries. It also has a complete tuple of first integrals, and the angle ψ is a cyclic coordinate. Theorem 8.2. System (175) is topologically equivalent to (154). Therefore, as in the plane case, there is a mechanical analogy between the pendulum in the medium flow and the free body in the presence of a certain nonintegrable constraint. Remark. The angle α for a free body is equivalent to the angle θ, whereas the angle β is equivalent to the angle ψ. Moreover, for systems (154) and (175) to be identical, it is necessary to set l = −σ and v∞ = v. The constant velocity of the characteristic point of the plane domain for a free body corresponds to the constant velocity of the over-running flow on the pendulum. The relation l = −σ tells us that for a free body, the stationary motion α ≡ 0 is exponentially unstable, and for the pendulum, the stationary motion θ ≡ 0 is exponentially stable. 859
8.4.3. Remarks on a dynamically symmetric rigid body in a special force field. System (154) has a variable dissipation and is not conservative. It becomes conservative by formally equating the quantity σ to zero. This means that the plane plate moves in a medium some point of which has a constant velocity. One more analogy with the Lagrange rigid body holds in a special force field. In the case where the longitudinal component of the angular velocity is equal to zero, let the following force act on the Lagrange rigid body: it is perpendicular to the equatorial plane, and its value is equal to C1 s(θ),
C1 > 0
(176)
(θ is the nutation angle), and the distance from the application point to the dynamical symmetry axis is equal to (177) C2 R(θ), C2 > 0. Then the dynamical equations of motion take the form F(θ) 2 Ω˙y = − v cos φ, I2 F(θ) 2 Ω˙z = v sin φ. I2
(178)
Here φ is the angle of the proper rotation and v = const. Then by the Euler kinematic formulas, introducing the changes z1 = −Ωz cos φ − Ωy sin φ,
z2 = Ωz sin φ − Ωy sin φ,
(179)
we have θ˙ = −z2 , z˙2 = n20 v2 sin θ cos θ − z21
cos θ , sin θ
cos θ , sin θ cos θ . φ˙ = z1 sin θ
(180)
z˙1 = z1 z2
(181)
Corollary 8.1. The system (180), (181) is equivalent to (154) , (155). Therefore, we have the following three model problems equivalent to each other: (1) a free rigid body in the presence of a certain nonintegrable constraint; (2) a pendulum in the medium flow; (3) a dynamically symmetric rigid body in a special force field. 8.5. Topological Structure of Phase Portrait of System Studied. In this subsection, we present a scheme of global qualitative analysis of the dynamical system (151) on the whole phase space R3 {α, z2 , z1 }. For any function F ∈ Φ, the phase portrait of system (151) has the same topological type. System (151) has no trajectories having infinitely distant points of the phase space as its α- and ω-limit sets. Moreover, the system has no simple and complicated limit cycles. 8.5.1. Reducing the system to the form studied. For convenience of drawing the three-dimensional phase portrait and preserving the right-oriented coordinate system {α, z2 , z1 }, let us make the formal change α → −α. Moreover, because of the existence of symmetries, we study the domain {(α, z2 , z1 ) ∈ R3 : − π < α < 0, z1 > 0}. 860
(182)
In this case, the system takes the form v F(α) , I2 cos α F(α) 2 cos α z˙2 = − v + z21 , I2 sin α cos α , z˙1 = −z1 z2 sin α and under condition (16), the analytic system has the form α˙ = z2 + σ
(183)
α˙ = z2 + σn20 v sin α, z˙2 = −n20 v2 sin α cos α + z21
cos α , sin α
(184)
cos α z˙1 = −z1 z2 . sin α For simplicity, let us study system (184), and let us “rectify” the field along the cylinders {(α, z2 , z1 ) ∈ R3 : z2 + σn20 v sin α = 0}; precisely, making the change of the phase variables u = z2 + σn20 v sin α, we pass from system (184) to the system α˙ = u, cos α + σn20 vu cos α, (185) sin α cos α . z˙1 = −z1 [u − σn20 v sin α] sin α For σ = 0, system (185) (denoted by (185) ) has two analytical integrals. The following proposition is obvious. u˙ = −n20 v2 sin α cos α + z21
Proposition 8.3. The plane {(α, u, z1 ) ∈ R3 : z1 = 0}
(186)
is integral for system (183). Corollary 8.2. Plane (186) “contains” the portrait of the system from the plane dynamics (see Fig. 3 if we extend the system field to the lines {(α, u) ∈ R2 : sin α = 0} by continuity). Let us introduce the family of (three-dimensional) layers Π(α1 ,α2 ) = {(α, x1 , x2 ) ∈ R3 : α1 < α < α2 }.
(187)
8.5.2. Conservative third-order comparison system. In Sec. 4, we have presented many assertions concerning PTS’ and more general comparison systems on two-dimensional manifolds. To study third-order systems, we need PTS’ and comparison systems of higher dimension. We do not dwell on the general theory of PTS’ and comparison systems of higher dimensions and restrict ourselves to its application to the system studied. Since system (185) has two analytical integrals (160), the latter fiber the phase space at each point of which we can draw two surfaces given by relations (160) that intersect along the phase characteristic of system (185) . For each point of the phase space of system (185), let us define two pairs of subspaces in each of which the characteristic of system (185) enters or emanates from. The first integrals (160) “help us” to study the behavior of phase trajectories of system (185). If g1 = {n20 v2 sin α cos α, u, z1 }, g2 = {z1 cos α, 0, sin α} are the gradients of surfaces (160), then the inner products χk = (gk , v), k = 1, 2 (v is the vector field of system (185)) have all the properties of the characteristic functions (see Sec. 4). 861
Proposition 8.4. The characteristic functions χk have the form χ1 (α, u, z1 ) = σn20 v cos α[u2 + z21 ],
χ2 (α, z1 ) = σn20 vz1 sin α cos α.
8.5.3. Equilibrium points of system studied. The system (185) under study has the following equilibrium states: (a) the repelling point (0, 0, 0); (b) the attracting 0, 0); point (−π, (c) the saddles − π2 , 0, Q1 and Q1 ∈ [0, n0 ) in each small area parallel to the plane Oαz2 ; (d) the centers − π2 , 0, Q2 and Q2 ∈ (n0 , +∞) in each small area parallel to the plane Oαz2 . In the cases (c) and (d), the singular points are not isolated. Through the whole section, we have performed the qualitative analysis in sufficient detail. The next subsection is a consequence of the previous material. 8.5.4. Phase portrait structure. Proposition 8.4 implies the following assertions. (1) The point (a) is the α-limit set of the separatrices entering the points (c) in the layer Π(−π/2,0) . (2) The point (b) is the ω-limit set of the separatrices emanating from points (c) in the layer Π(−π,−π/2) . (3) The ω-limit (α-limit) sets of the separatrices emanating from (entering) the points (c) in the layer Π(−π/2,0) (in the layer Π(−π,−π/2) ) are the same points. (4) The part of the phase space containing points (d) is entirely filled with closed trajectories. 8.6. Trajectories of Spherical Pendulum Motion and the Case of Its Nonzero Twist near the Longitudinal Axis. 8.6.1. Pendulum trajectories on sphere. In accordance with the properties of the phase space partition into trajectories, the typical trajectories of the point D of the plane domain fall into classes. (1) The trajectories corresponding to the oscillatory domain. Such trajectories are curves on the sphere that unboundedly approach the poles of the sphere (along the flow) as t → ±∞. (2) The trajectories corresponding to the rotational domain. Such trajectories are curves that almost always densely fill annulus-like domains on the sphere symmetric with respect to the equator. 8.6.2. Spherical pendulum under nonzero proper twist. We immediately present the equations of the pendulum motion under the condition Ωx0 = 0. These equations have the following form analogous to (175): I1 sin θ sin θ 2 − Ωx0 ψ˙ = 0, sin θ cos θ − ψ˙ 2 θ¨ + ln20 v∞ θ˙ cos θ + n20 v∞ cos θ I2 cos θ (188) I1 cos θ 1 + cos2 θ 2 ˙ ˙ ¨ ˙ ˙ = 0. ψ + θψ + ln0 v∞ ψ cos θ + Ωx0 θ cos θ sin θ I2 sin θ Now let us immediately pass to the classification of possible pendulum trajectories on the sphere. (1) Trajectories analogous to trajectories a) for the case Ωx0 = 0. The asymptotics of behavior of such curves is the same as above. (2) The trajectories analogous to trajectories b) for the case Ωx0 = 0. Such trajectories are everywhere dense on the whole sphere [346]. 9. Methods for Analyzing Nonzero Mean Variable Dissipation Dynamical Systems in Rigid Body Spatial Dynamics This section (similar to Sec. 7) is devoted to the study of the most interesting (in application) class of rigid body motions, a free drag in a resisting medium. It is in fact an introduction to the problem of spatial free drag. Here we obtain some particular solutions of the complete system and prepare the material for performing the qualitative integration of the dynamical equations in the quasi-velocity space. The second part of the section is devoted to a new two-parameter family of phase portraits in the three-dimensional 862
space. We show that the obtained family consists of an infinite set of nonequivalent phase portraits with different nonlinear qualitative properties. Consider the class of motions under which a body executes a spatial free drag in a resisting medium. We can transform system (141) into the following form (zi = zi v, i = 1, 2; α˙ = α v; β˙ = β v; v˙ = v v): I1 v = Ψ1 (α, Z1 , Z2 )v + σ Ωx0 Z1 sin α, (189) I2 σ σI1 s(α) α = −Z2 + σ(Z12 + Z22 ) sin α + F(α) cos α + sin α, (190) Ω 0 Z1 cos α + I2 vI2 x m
1 σI1 Z2 = F(α) + Z2 −Ψ1 (α, Z1 , Z2 ) − Ω 0 Z1 sin α − Z1 Ψ2 (v, α, Z1 , Z2 ), (191) I2 vI2 x
σI1 Ω 0 Z1 sin α + Z2 Ψ2 (v, α, Z1 , Z2 ), (192) Z1 = Z1 −Ψ1 (α, Z1 , Z2 ) − vI2 x cos α Ωx0 Z2 σI1 Ωx0 − + , (193) β = Z1 sin α v vI2 sin α where σ s(α) cos α, Ψ1 (α, Z1 , Z2 ) = −σ(Z12 + Z22 ) cos α + F(α) sin α − I2 m I1 Ωx0 cos α σI1 Z2 + Z1 + . Ωx0 Ψ2 (v, α, Z1 , Z2 ) = − I2 v sin α vI2 sin α Equations (189)–(192) compose a closed fourth-order subsystem, and for Ωx0 = 0, Eqs. (190)–(192) compose a third-order subsystem. In this case, we assume that the functions F and s entering the right-hand side of system (189)–(192) satisfy conditions (15) and (13). In particular, if conditions (16) and (14) hold, then the systems (189)–(192) takes the form I1 (194) v = Ψ1 (α, Z1 , Z2 )v + σ Ωx0 Z1 sin α, I2 σI1 B (195) Ωx0 Z1 cos α + sin α cos α, α = −Z2 + σ(Z12 + Z22 ) sin α + σn20 sin α cos2 α + vI2 m
σI1 2 Z2 = n0 sin α cos α + Z2 −Ψ1 (α, Z1 , Z2 ) − Ω 0 Z1 sin α − Z1 Ψ2 (v, α, Z1 , Z2 ), (196) vI2 x
σI1 Ω 0 Z1 sin α + Z2 Ψ2 (v, α, Z1 , Z2 ), (197) Z1 = Z1 −Ψ1 (α, Z1 , Z2 ) − vI2 x where Ψ1 (α, Z1 , Z2 ) = −σ(Z12 + Z22 ) cos α + σn20 sin2 α cos α −
B cos2 α, m
I1 Ωx0 cos α σI1 Z2 Ωx0 + Z1 + . I2 v sin α vI2 sin α For Ωx0 = 0 (the case of absence of body self-rotation) the systems (189)–(192) and (194)–(197) take the following form respectively: Ψ2 (v, α, Z1 , Z2 ) = −
v = Ψ1 (α, Z1 , Z2 )v, σ s(α) sin α, F(α) cos α + I2 m 1 cos α Z2 = F(α) − Z2 Ψ1 (α, Z1 , Z2 ) − Z12 , I2 sin α cos α Z1 = −Z1 Ψ1 (α, Z1 , Z2 ) + Z1 Z2 , sin α α = −Z2 + σ(Z12 + Z22 ) sin α +
(198) (199) (200) (201) 863
where Ψ1 (α, Z1 , Z2 ) = −σ(Z12 + Z22 ) cos α +
σ s(α) cos α, F(α) sin α − I2 m
and v = Ψ1 (α, Z1 , Z2 )v,
(202)
α = −Z2 + σ(Z12 + Z22 ) sin α + σn20 sin α cos2 α + Z2 = n20 sin α cos α − Z2 Ψ1 (α, Z1 , Z2 ) − Z12 Z1 = −Z1 Ψ1 (α, Z1 , Z2 ) + Z1 Z2
cos α , sin α
cos α , sin α
B sin α cos α, m
(203) (204) (205)
where
B cos2 α. m Under certain natural conditions, the systems (194)–(197) and (202)–(205) reflect the main topological properties of the partition of the phase space R1+ {v} × R3 {α, Z1 , Z2 } into trajectories of the systems (189)–(192) and (198)–(201). Moreover, the vector fields of the systems (189)–(192) and (194)–(197) (resp. (198)–(201) and (202)–(205)) are topologically equivalent. Ψ1 (α, Z1 , Z2 ) = −σ(Z12 + Z22 ) cos α + σn20 sin2 α cos α −
9.1. Case of Zero Longitudinal Component of Angular Velocity and Corresponding Stationary Motions. Let us consider the case where Ωx0 = 0. From the initial system (198)–(201), the third-order subsystem (199)–(201) is separated. Therefore, in the phase space of the latter, the equilibrium points can be the projection of nonsingular phase trajectories of the four-dimensional phase space. Indeed the system (198)–(201) has equilibrium states that fill one-dimensional manifolds. Therefore, the problem on the equilibrium points falls into the following two problems: the problem on equilibrium points of the system (198)–(201) in the space R1+ {v} × R3 {α, Z1 , Z2 } and that on the equilibrium points of the truncated system (199)–(201) in the space R3 {α, Z1 , Z2 }. 9.1.1. Equilibrium points of fourth-order system. The desired equilibrium points are given by the following system with positive parameter v1 : 1 2 1 π Z2 − + Z12 = 2 , v = v1 . (206) α= , 2 2σ 4σ In the phase space R1+ {v} × R3 {α, Z1 , Z2 }, system (206) defines a one-dimensional manifold (circle) entirely filled with the equilibrium points of the system (198)–(201). 9.1.2. Mechanical interpretation of equilibrium points of four-dimensional phase space. In contrast to the plane case, the equilibrium points (206) of the four-dimensional phase space have a richer mechanical interpretation. Therefore, let us consider the following particular solution of the system (198)–(201): π α = , v = v0 , β = 0, Ωy = Ωy0 , Ωz = Ωz0 . 2 The kinematic relations Ωy0 sin φ, θ˙ = − cos ψ (207) ψ˙ = Ωy0 cos φ, sin ψ φ˙ = Ωz0 + Ωy0 sin φ cos ψ admit the “separation” of the independent second-order subsystem (207), which has an analytical first integral. 864
Proposition 9.1. System (197) has a first integral of the form Ωy0 sin φ cos ψ − Ωz0 sin ψ = C1 = const,
(208)
which means that the projection of the absolute angular velocity on the axis Ox0 is preserved (see Sec. 8.1). By Proposition 9.1 and the first equation of the second-order system (207), the angle ψ is found from the relation Ωy0 (Ω2y0 + Ω2z0 ) − C12 C1 Ωz0 2 + Ω2 )(t + C ) − sin (Ω , (209) sin ψ = ± 2 0 0 2 2 2 y z Ωy0 + Ωz0 Ωy0 + Ω2z0 where C1 = const. Identities (208) and (209) allow us to find the dependence of the angle φ on time through elementary functions. The following relation holds on the phase trajectories studied: v0 [−C1 − Ωz0 sin ψ] x˙0 = −v0 cos ψ sin φ = Ωy0 and by (209),
v0 [A1 ∓ A2 sin(A3 t + A3 )]. Ωy0 Therefore, for C1 = 0 (plane case, Ωz0 = 1/σ, α = π/2), the coordinate x0 periodically varies in time. Analogously, we find y0 and z0 . Despite the finding of only one Cartesian coordinate, we can conclude that the body motion is stationary. Let Z1 = Z2 = 0 (at the initial instant, the plane case). In this case, the body executes a translational motion at constant velocity parallel to the plane domain. According to Proposition 9.1, the axis Ox0 is directed parallel to the plane domain (for any Z1 and Z2 in the process of motion). If we fix a constant velocity of translational motion for the latter initial conditions and assign an initial shock in the direction perpendicular to the axis Ox0 , then the characteristic point D of the plane domain (see Fig. 46) executes a screw motion under which the body center of mass moves rectilinearly and uniformly, whereas the rotational motion is executed at a constant velocity. Increasing the angular velocity of the body rotation and diminishing the velocity of the center of mass down to zero, from the latter initial conditions, we obtain the point given by the relations Z2 = 1/σ and Z1 = 0, which also corresponds to motion from the plane dynamics. x˙0 =
9.1.3. Trivial and nontrivial equilibrium points of third-order system. The system (199)–(201) has equilibrium points in the space R3 {α, Z1 , Z2 }, which are given by the relations: (210) α = πk, k ∈ {0, 1}, Z1 = Z2 = 0, 2 1 1 π Z2 − + Z12 = 2 . (211) α= , 2 2σ 4σ As was shown above, the equilibrium points of the truncated system can be the orthogonal projections either of nonsingular phase trajectories or of manifolds entirely filled with the equilibrium points of the phase space R1+ {v} × R3 {α, Z1 , Z2 }. In the phase space, system (211) defines the equilibrium points into which the manifolds of equilibrium states are orthogonally projected. Conversely, the manifolds of equilibrium states are projected only into the points defined by system (211). Hence system (210) defines the points into which particular solutions of the form B
v(q) = v0 e− m q , α(q) ≡ πk,
v0 = v(0),
k ∈ {0, 1},
(212)
Z1 (q) ≡ Z2 (q) ≡ 0 are orthogonally projected. 865
The mechanical interpretation of particular solutions (212) is as follow: the body executes a rectilinear translational motion with the angle of attack equal to 0 or π. In this case, the velocities of all points of the body decay in time as t−1 . Because of the simplicity of the mechanical interpretation for the stationary motions of the equilibrium points (210) and (211), the latter are called the trivial equilibrium states (analogously to the plane dynamics). Along with the trivial equilibrium states, there can exist equilibrium points corresponding to the nontrivial particular solutions of the system. Definition. The nontrivial equilibrium states (NES’) of the system (199)–(201) in the space R3 {α, Z1 , Z2 } are equilibrium points not lying on the planes {(α, ω) ∈ R2 : Z2 sin α cos α = 0}. 9.1.4. Nontrivial equilibrium states of third-order system corresponding to nontrivial particular solutions. Let us introduce the notation n20 = F (0)/I2 (as in the previous sections). For simplicity, let us consider the system (203)–(205). The system (199)–(201) can be considered analogously. The necessary and sufficient condition for the existence of NES’ in the space not lying on the integral plane {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0} is the expression of these equilibrium state by the system σn20 σn2 m σ 2 n20 2 2 sin α; Z1 = n0 1 − Z2 = sin2 α, σn0 < 2; cos α = − 0 , 2 4 2B and, moreover, outside the plane {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0}, NES’ appear only for σn0 < 2, and for σn0 > 2, there are none of them at all. Therefore, for σn0 < 2, on the plane {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0} , there are no NES’ (see the plane dynamics). For σn0 ≥ 2, there are no NES’ in the space {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0}. For σn0 < 2, NES’ on the plane {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0} “disappear” and “move” to the space {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0}. Corollary 9.1. Under the conditions (15), (13), and Ωx0 = 0, the system (198)–(201) has particular solutions of the form v(q) = v0 eκq ,
v = v0 (0), κ < 0; cos α0 q + β 0 , β 0 = β(0); α(q) ≡ α0 , α0 = α(0); β(q) = Z10 sin α0 Z1 (q) ≡ Z10 , Z2 (q) = Z20 , Zi0 = Zi (0), i = 1, 2. Corollary 9.2. Under the conditions (15), (13), and Ωz0 = 0, the system (141) has particular solutions of the form v0 , v0 = v(0); v(t) = 1 − v0 κt β α(t) ≡ α0 , α0 = α(0); β(t) = β0 − 0 ln(1 − v0 κt); v0 κ and also 1 [Ω 0 cos β(t) − Ωz0 sin β(t)], Ωy (t) = 1 − v0 κt y 1 Ωz (t) = [Ω 0 sin β(t) + Ωz0 cos β(t)] 1 − v0 κt y if β0 = 0 analogous to the corresponding solutions from the plane dynamics. 9.2. Bundles of Phase Space, Its Symmetries, and Beginning of Topological Analysis. 9.2.1. Topological classification of singular points of truncated system in three-dimensional space. As previously, let us introduce the following two dimensionless parameters: B , µ2 = σn0 . µ1 = 2 mn0 Let us consider the system (198)–(201) for σn0 < 2. 866
Fig. 48: One-dimensional manifold of equilibrium states.
Proposition 9.2. 1 , in each small area perpendicular to the circle, the nonisolated equilibrium points (211) (1) For Z2 < 2σ are saddles (Fig. 48). 1 , in each small area perpendicular to the circle, the nonisolated equilibrium points (211) (2) For Z2 > 2σ are attracting. (3) For k = 0, the isolated equilibrium point (210) is always repelling (Fig. 49). (4) For k = 1, the isolated equilibrium point (210) is repelling if µ2 < µ1 and attracting if µ2 > µ1 (Fig. 49).
9.2.2. Bundles of phase space and its symmetries. Since an independent third-order subsystem is separated from the fourth-order system, the phase trajectories in R1+ {v} × R3 {α, Z1 , Z2 } lie on surfaces that are three-dimensional cylinders. In particular, if in the whole phase space, there exists an additional first integral of the system (199)–(202), then it is a function of the variables (α, Z1 , Z2 ), and, therefore, it defines a family of cylinders in R1+ {v} × R3 {α, Z1 , Z2 }.
Fig. 49: Attracting and repelling singular points.
867
Owing to the reduction described above, it is more convenient to construct the phase portrait of the system (198)–(201) in R1+ {v} × R3 {α, Z1 , Z2 } by using the phase portrait of the system (199)–(201) in R3 {α, Z1 , Z2 }. The phase portrait in the space is not a part of the phase portrait in R1+ {v} × R3 {α, Z1 , Z2 } in the set-theoretic sense, but it is the projection of the phase portrait on the plane {v = const}. In this case, the projection is orthogonal. Owing to the cylindrical nature of the vector field, it becomes possible to lift the phase trajectories to the space R1+ {v} × R3 {α, Z1 , Z2 } and to obtain the four-dimensional phase portraits of the desired system. These assertions can be generalized to the case of higher dimensions. Since v > 0, motion is possible only in the domain B = {(α, Z1 , Z2 , v) ∈ R4 : v > 0} ⊂ R4 {α, Z1 , Z2 , v}. If we formally make the change of variables in the domain B by the formula p¯ = ln v, then the obtained vector field in the phase space R4 {α, Z1 , Z2 , p¯} is independent of p¯, i.e., it has a cylindrical nature of higher order, and, simultaneously, it is uniquely orthogonally projected on the whole family of planes {¯ p = const}. In this case, the equilibrium states in the space R3 {α, Z1 , Z2 } completely coincide with the union of the projections of manifolds of singular points, as well as with the union of the projections of nonsingular phase trajectories of domain B. To solve the problem of lifting the trajectories to the four-dimensional phase space completely, it is necessary to reveal the sign of the projection of the vector field in R4 {α, Z1 , Z2 , p¯} on the axis p¯ or on the axis v in the domain B. For any F ∈ Φ and s ∈ Σ, the vector field of the system (199)–(201) has the central symmetry property with respect to the points (πk, 0), k ∈ {0, 1}, i.e., in the coordinates (α, Z1 , Z2 ), the vector field changes its directions under the change πk − α πk + α → , k ∈ {0, 1}. −Z1 , −Z2 Z1 , Z2 Moreover, the plane {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0} is integral, whereas the vector field of the system has the following symmetry: the α- and Z2 -components are preserved, whereas the Z1 -component changes its sign under the change α α → , k ∈ {0, 1}. Z1 , Z2 −Z1 , −Z2 For the further analysis, in the space R3 {α, Z1 , Z2 }, let us introduce the family of layers Π(α1 ,α2 ) = {α, Z1 , Z2 ) ∈ R3 : α1 < α < α2 }; in this case, as in the previous sections, Π(− π2 , π2 ) = Π,
Π( π , 3π ) = Π . 2
2
It should be noted that the phase space of the system studied is in fact the layer Π(0,π) (or, topologically, the tangent bundle of the two-dimensional sphere). 9.3. Classification of System Phase Portraits in Three-Dimensional Space for a Certain Parameter Domain. 9.3.1. Introduction to classification of phase portraits. Similar to the plane dynamics, we can depict the surgery diagram of the vector field of the system (199)–(201) near the point (π, 0) on the set Π(0,π) ∪ {(α, Z1 , Z2 ) ∈ R3 : Z1 > 0}. The behavior of the system vector field determines many main topological features of the global phase portrait. 868
We study those dynamical systems of the form (199)–(201) for which NES’ exist only outside the integral plane {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0}. Therefore, in the general space of physically admissible parameters M2 = {(µ1 , µ2 ) ∈ R2 : µ1 > 0, µ2 > 0}, we mainly study the domain J2 = {(µ1 , µ2 ) ∈ R2 : µ1 > 0, 0 < µ2 < 2} ⊂ M2 . The typical topological classification of trivial equilibrium states was carried out above. To perform the complete topological classification of the system phase portraits, it is necessary to answer the following questions. (1) The existence and uniqueness of trajectories that have infinitely distant points as their α- and ω-limit sets. Proposition 9.3. The system (199)–(201) has unique trajectories going to infinity. Their α- and ω-limit sets are the infinitely distant points (+0, +∞, +∞) outside the plane {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0} and (+0, 0, +∞) on the plane {(α, Z1 , Z2 ) ∈ R3 : Z1 = 0}. Only these points have the above property. (2) The existence of closed curves of trajectories contractible to a point along the layer Π(0,π) . In the general statement, the latter problem is very complicated, but in the domain of the parameters considered, the following proposition holds. Proposition 9.4. In the parameter domain I (cf. the results of the plane dynamics; see Sec. 7), under conditions (15) and (13), systems of the form (199)–(201) have no closed characteristics. (3) The main problem in the classification of portraits is the problem on the behavior of stable and unstable separatrices that the existing hyperbolic saddles have. These separatrices divide the whole phase space into domains without equilibrium states, and, therefore, the phase portraits are instantaneously completed. The answer to the latter question requires consideration of a number of auxiliary assertions. 9.3.2. Partition of phase space into trajectories of systems for first parameter domain. Let us consider the system (199)–(201) in the parameter domain I (see Sec. 7). We study the behavior of stable and unstable separatrices that hyperbolic saddles have. The following assertion refers to the whole parameter space M2 and is based on the existence of a three-dimensional Poincar´e topografical system in the strip Π. Proposition 9.5. For Z2 < as their α-limit sets.
1 2σ ,
the unstable separatrices in the layer Π for points (211) have the origin
Proposition 9.6. Let us consider the system (199)–(201) in the parameter domain I. Let F(α) sin α ≥ 0 in the strip Π. Then if σ s(α) ≥ gˆ(α), m I where π gˆ (α) = −2F(α), gˆ = 0, 2 it follows that the PTS centered at the point (π, 0) entirely lies in the domain bounded by a nontrivial contact surface (see an analogous Proposition 4.2 for a nontrivial contact curve on the plane). 869
This proposition is formulated and proved analogously to the proposition from the plane rigid body dynamics with the one exception that the nontrivial contact surface transferred to the strip Π has the form
s(α) F(α) s2 (α) s(α) σF(α) + 4σ + sin α − . 2σ(Z12 + Z22 ) = m m2 I m cos α I sin α Corollary 9.3. Let condition (16) hold. Then, obviously, in the parameter domain I, the following inequality holds: π π B ≥ σn20 cos α for all α ∈ − , . m 2 2 Moreover, gˆ(α) = AB cos2 α. Therefore, for F = F0 , we are necessarily in the conditions of Proposition 9.6. Proposition 9.7. Let all conditions of the previous propositions hold. Then at least in the parameter 1 in the layer Π have the point (π, 0, 0) as domain I the separatrices entering the points (211) for Z2 < 2σ their α-limit set. 9.3.3. Topological structure of phase portrait types for first parameter domain. Let us consider the problem 1 to the strip Π and the strip Π . on the behavior of the separatrices emanating from points (211) for Z2 < 2σ For this purpose, let us give the definition of the index of separatrix behavior for this problem. Definition. The index of separatrix behavior (denoted by isp) is the number
1 k ∈ N0 ∪ 1 + , l ∈ N0 . 4 1 By definition, isp = k if the separatrices emanating from points (211) ! for Z2 < 2σ in" the layer Π have 1 1 points (211) for Z2 > 2σ (k ∈ N0 ) or the infinitely distant point (k ∈ l + 4 , l ∈ N0 ) as their ω-limit sets. Moreover, the separatrices envelope circle (211) and go to the domain under circle (211) l times (Fig. 50 for l = 0 and Fig. 51 for l = 2).
Fig. 50: l = 0.
870
Fig. 51: l = 2.
Theorem 9.1. The definition is correct, i.e., for any isp = k from its domain, the corresponding behavior of separatrices is possible. Theorem 9.1 is proved by the methods of monotone vector field theory (see Sec. 4). The monotonicity is understood with respect to the parameter s∗ , s∗ = max |s(α)| µ∗ = 2 α∈R mn0 (see [279]; cf. analogous results from the plane dynamics in Sec. 7). 9.3.4. Complete classification of phase portraits of dynamical system in three-dimensional space for first parameter domain. By Theorem 9.1, we can perform the complete classification of phase portraits of the system (199)–(201) when its parameters vary in the domain I. As was shown above, there exist infinitely many such portraits. To perform the complete classification of portraits, it remains to study the separatrices emanating 1 to the layer Π . Such separatrices have limit sets consisting of NES’. The from points (211) for Z2 < 2σ originating singular point (NES) in the half-space {Z1 > 0} is of the saddle type with one attracting eigendirection and two repelling eigendirections. It is noteworthy how the points from NES’ introduce possible changes in the properties of the partition of the three-dimensional phase space into trajectories. The stable eigendirection (points from NES’) has the point (π, 0, 0) as its α-limit set. The unstable directions on which the whole “unstable” plane is spanned (and also the separatrices about which we speak in the latter definition) have the attracting 1 . It is seen that the saddle points from NES’ introduce points (211) as their limit sets for Z2 > 2σ changes in the topological type of the three-dimensional phase portrait that have no influence on it. The topological types of three-dimensional phase portraits “are coded” only by the index k = isp, which “does not control” the other separatrix surfaces. In this way, the complete topological classification of the phase portraits of the system is performed for the parameter domain I. Note that many assertions of this subsection also hold in wider parameter domains. In conclusion, we note a certain general property of the body motion. Let us consider the parameter domain I. The dimensionless moment, which is small compared with the dimensionless force, corresponds to it. In this domain, for typical (in the sense of measure theory) initial conditions, in a sufficiently long time, the body tends to the exponentially stable stationary motion of the following form: the body center 871
of mass moves rectilinearly and uniformly, whereas the body rotates around the center of mass at constant angular velocity in the direction perpendicular to the velocity of the center-of-mass motion. In this case, the velocity of the relative motion under the rotation is greater than the (translational) velocity of the center of mass. In the material just presented, we have considered a model variant of the problem of free spatial drag of a body in a medium. We have carried out the auxiliary qualitative analysis of the systems of differential equations describing this motion in a certain domain of nonzero measure in the parameter space. Basing on this, we have obtained a new two-parameter family of phase portraits consisting of an infinite set of nonequivalent topological types. In this case, there are no auto-oscillations, and for typical initial conditions, the trajectories tend to asymptotically stable stationary regimes. 10. Some Problems of Plane Dynamics of a Rigid Body Interacting with a Medium in the Presence of Linear Medium Damping Up to now, we have considered the nonlinear model of interaction of a body with a medium in which there is no medium damping (h = 0 in the linear case; see Sec. 1) and the resistance force and its moment depend only on the angle of attack. In this section, we pass to the account of an additional linear medium damping action. On the basis of nonlinear equations, we study the stability of the rectilinear translational drag in the presence of a linear damping moment. We show that in the framework of the model considered in principle there can arise auto-oscillations corresponding to limit cycles that are born from a weak focus (the well-known Andronov–Hopf bifurcation). Also, we perform the qualitative analysis of some nonlinear dynamical systems obtained above but under the condition that there exists a linear damping moment in the system considered. Depending on the medium damping coefficient, we perform the topological classification of typical phase portraits of the system considered on the quasi-velocity phase cylinder. We show that under certain conditions, there can arise stable as well as unstable auto-oscillations in the system. The results obtained in the previous sections are based mainly on the principal condition on the pair (yN , s) of dynamical functions, their dependence only on the angle of attack (see also [381, 382, 407, 408, 412]). The present section is devoted to studying the influence of the rotational derivative of aerodynamic force moment in the angular velocity on the body motion itself. In the general case, the account of such an influence meets serious difficulties related to a complicated dependence of the aerodynamic force moment on the body angular velocity. First we study the case of the linear dependence of yN on the body angular velocity with coefficient constant in the angle of attack. 10.1. Body Free Drag in a Medium Taking Account of Linear Damping Moment. 10.1.1. Exponential stability conditions of body rectilinear translational drag. Let us consider the problem of the body free drag in a medium. Assume that the ordinate of the point of application of the resistance force is represented in the form yN (α) − h Ωv . In this case, Eq. (10) takes the form IΩ = F(α)v2 − h s(α)Ωv,
(213)
where h is a constant coefficient. The corresponding dynamical equations analogous to (86), (87) take the form v = vΨ(α, ω), σ σ s(α) F(α) cos α + σω 2 sin α + sin α − h ωs(α) cos α, I m I h 1 ω = F(α) − ωΨ(α, ω) − ω, I I
α = −ω +
872
(214) (215) (216)
where
s(α) σ σ F(α) sin α − cos α − h ωs(α) sin α. I m I As is known, for h = 0, the body rectilinear translational drag is exponentially unstable in a part of the variables (α, Ω), i.e., the trivial solution of the independent subsystem (215), (216) is exponentially unstable. In studying the body rectilinear translational drag, from the experiment at the entrance of the circular cylinder into the water (see Secs. 1 and 3), it was obtained that the coefficient h of the medium action on the body is strictly positive, i.e., it brings damping in the system. Therefore, first we consider the case where h1 = h s(0) > 0. Ψ(α, ω) = −σω 2 cos α +
Proposition 10.1. If
F (0) s2 (0) B2 n20 = > = , I I m2 m2 then there exists a critical value B ∗ 2 h1 = I σn0 + 2 , m such that for h1 > h∗1 , the trivial solution of the system (215), (216) is exponentially stable.
(217)
Let us consider the problem of the entrance of circular cylinders into the water (see Secs. 1 and 3). The condition for the existence of the oscillatory exponential stability (217) takes the form 8k DIρl , (218) < 2 m Cx π and the values of the cylinder physical parameters under which the rectilinear translational drag can be stable in principle must satisfy the condition mσD mD2 >2+k . (219) I I We have already assumed that the dimensionless parameters k and h of the water action on the body satisfy the estimate k = h = 0.1 (see Sec. 1). Therefore, conditions (218) and (219) allow us to try to “construct” a rigid body, a circular cylinder, for which the rectilinear translational drag can be stable. For this purpose, it is necessary to choose the dynamical parameters σ, D, I, and m of the cylinder proceeding from conditions (218) and (219). Initially, the latter fact contradicts the assertion that the rectilinear translational drag of homogeneous circular cylinders in the water is unstable. The dynamical parameters of the cylinders themselves are such that the inequality (218) holds, whereas (219) violates. Indeed, the following relation holds for circular cylinders moving in water (k = h = 0.1) with uniform mass distribution: mσD mD2 −k − 2 = F2 (σ, D); F1 (k, h, m, I, σ, D)|k=h=0.1 = h I I h
k=h=0.1
with accuracy up to a positive factor, the right-hand side of this relation assumes the negative value −3D2 − 12σD − 80σ 2 , which corresponds to the exponential instability of the rectilinear translational drag; here, we take into account that 2 D2 σ + . I=m 3 16 Let us trace the formal dependence of the function F1 on the parameters k and h, which are seemingly determined by the medium itself. It is possible that the parameters k and h grow when the medium viscosity grows. In this case, for √ 2 5−4 ∼ k=h> = 0.157, 3 873
there exist dynamical parameters of homogeneous circular cylinders for which the function F1 becomes positive, which testifies to the possible stability of the rectilinear translational drag. 10.1.2. On motion of circular cylinders with a hole. As was shown above, the rectilinear translational drag of homogeneous circular cylinders is unstable. For the possible attainment of such a stability, let us consider the problem of a free drag of a circular cylinder having a hole inside it. In this case, the moment of inertia of the cylinder with a hole with respect to the geometric cylinder center of symmetry is equal to 3 D2 σ 2 + −ρ , I=m 3 16 where ρ is the central radius of inertia of a conceptual rigid body filling the hole. Then the stability condition of the rectilinear drag of the body considered takes the form ρ 2 3 3 σ σ 2 3 + + > . D 80 20 D D For example, if the cylindrical hole in the center of the “ambient” cylinder is of diameter D1 = k1 D and the half-height σ1 = k2 σ (0 < k1 , k2 < 1), then for σ 2 1 σ 3 3 5k12 − 1 + < 0, − D 20 1 − k22 D 80 1 − k22 the rectilinear translational drag is exponentially stable. 10.1.3. Appearance of auto-oscillations by Andronov–Hopf cycle birth bifurcation. Let us consider the problem of stability of the trivial solution to the system (215), (216) at the boundary of the stability region subject to the critical value of the medium damping parameter h∗1 . Introduce the following notation analogous to that of Sec. 3, precisely: s2 f3 B B 2 2 2 2 , In = −2 + 4 + σ − 5σn0 + 8σA1 + 8σω0 + 4σA1 σn0 + 2 m m I m F (0) , s2 = s (0), f3 = F (0), B = s(0), n20 = I A1 =
σn20 +
B m
B 1 + σ 2 n20 + 2σ m
,
ω0 =
n20 −
B2 m2
B 1 + σ 2 n20 + 2σ m
.
Proposition 10.2. If In < 0, then for h1 = h∗1 , the origin of the system (215), (216) is asymptotically stable, and if In < 0, it is unstable. The value of the index In allows us to study the stability of the weak focus [31] at the origin of the system (215), (216) and thus to make a conclusion on the appearance of auto-oscillations in the system. Proposition 10.3. Let the conditions of Proposition 10.1 hold. If In < 0, then there exists h1 such that for 0 < h1 < h1 < h∗1 , the system (215), (216) has a stable limit cycle in the strip Π, which grows when h1 decreases from h∗1 down to h1 . If In > 0, then there exists h1 such that for h1 > h1 > h∗1 , system (215), (216) has an unstable limit cycle in the strip Π, which grows when h1 increases from h∗1 up to h1 . The index In depends on higher derivatives of the dynamical functions F and s whose signs are unknown in general. If we assume that conditions (16) and (14) hold, then the index In can be negative, which testifies to the birth of a stable limit cycle for h1 < h∗1 . To verify the latter fact, it is necessary to perform an experiment on the interaction with water, e.g., of a homogeneous circular cylinder having geometric and dynamical parameters the values of the function F2 of which are maximally close to zero to the left. Therefore, in principle, the rigid body dynamical parameters can be such that despite the exponential instability of the rectilinear translational drag of the body, its angular oscillations near this drag can be 874
uniformly bounded with respect to the amplitude. In other words, for any sufficiently small perturbation of the rectilinear translational drag with respect to the angle of attack and the angular velocity, the angular oscillations are bounded in amplitude by stable angular auto-oscillations, which is an affirmative answer to the principal question of the nonlinear analysis posed in Sec. 1. 10.2. Motion in a Medium in the Presence of a Certain Nonintegrable Constraint and Linear Damping Moment. In Sec. 6, we have carried out a complete study of the dynamical system (72) on the phase cylinder, which is separated from the general system (8)–(10). System (72) is defined by using the function space Φ and has a phase portrait that does not change its topological type when the function F varies in the whole class Φ, and, therefore, the integration of the kinematic equations was carried out for system (73) of a particular form. 10.2.1. Medium moment linear in angular velocity. Let us consider the problem of the body motion in a medium under condition (73) (see Se. 6). Assume that along with the resistance force, the body is subjected to the action of a damping moment linear in the angular velocity that changes only the angular acceleration. Assume that under condition (16), the change of the body angle of attack and angular velocity is given by the system α = −Ω + A1 sin α,
Ω = A2 sin α cos α − h2 Ω.
(220)
For h2 > 0, this additional moment brings a dissipation to the system, and for h2 < 0, it amplifies the swing effect of body angular oscillations. In accordance with the experiment, we choose the positive sign of h2 as previously. The system of the particular form (73) preserves all topological features of the phase portrait structure for the general system (72). But system (220) has no such property relative to system (92) because of the following reason. By Lemma 4.1, the space of vector fields of systems having the form (92) falls into at least two classes in each of which the corresponding phase portrait has a definite type. In the process of studying system (92), we show for which functions F ∈ Φ the qualitative analysis is performed. 10.2.2. Equilibrium points and stationary motions corresponding to them. Let us begin the global qualitative analysis of the dynamical system (92) under condition (15) on the quasi-velocity phase plane. The set of equilibrium points of system (92) (or a system of the particular form (220)) to which stationary motions correspond falls into the following parts. In the strip Π (or Π ), there exist isolated singular points, which are found from the system {α = 0 mod π, Ω = 0}. These singular points correspond to body rectilinear translational motions with constant velocity perpendicular to the plane plate. System (92) has the singular points A1 h A1 h A2 , ± F arccos (221) ± arccos A2 h A2 if and only if A2 . A1 Under the latter condition, they appear only in the strip Π. In the strip Π (and only in it), there arise the points A2 A1 h A1 h , ± F arccos − π ± arccos − A2 h A2 0 A1 F (0)) and if F2 (0) + 3F2 (0) < 0 for the portrait of system (92) with another function F = F2 (α) ∈ Φ (a stable cycle is born for h < A1 F (0)). In this case, it is necessary to distinguish concretely the subclass of functions from the class Φ for which the topological type of phase portraits of systems defined by using such a subclass is studied. 876
10.3.1. Topological classification of system equilibrium points. Let us perform a typical topological classification of the singular points(0, 0) and (π, 0) of system (220) on the cylinder, since the relations for the constants A1 and A2 obtained below are different from analogous relations for system (92) by the replacement of A1 by A1 F (0) and A2 by A2 F (0). The characteristic roots of system (92) linearized near the point (0, 0) have the form 1 λ1,2 = (A1 − h2 ± [(h2 + A1 )2 + 4A2 ]1/2 ), 2 and the bifurcation set consists of the constants
A2 A1 , − , −A1 + 2 −A2 , −A1 − 2 −A2 . A1 For the point (π, 0), we have analogous expressions under the replacement of A1 by −A1 . The equilibrium points (221) and (222) are saddles for any A2 A2 . h∈ − , A1 A1 An analogous equation for system (92) differs from the previous by a nonzero multiplicative constant. For the further presentation, it is convenient to introduce the following notation and abbrevitions: A2 ± ± , η+ = A1 ± 2 −A2 , η− = −A1 ± 2 −A2 , ξ= A1 S is a saddle, SN is a stable node, UN is an unstable node, SF is a stable focus, UF is an unstable focus, SC is a stable limit cycle, US is an unstable limit cycle, and PP is a phase portrait. The classification of the equilibrium points (0, 0) and (π, 0) is presented in Tables 1–4 and corresponds to the following four cases. Case 1 (Table 1): − − + + < η+ < −A1 < 0 < A1 < η− < η+ < −ξ. ξ < η− Case 2 (Table 2): Case 3 (Table 3): Case 4 (Table 4):
− − + + < ξ < η+ < −A1 < 0 < A1 < η− < −ξ < η+ . η− − − + + < −A1 < ξ < η+ < 0 < η− < −ξ < A1 < η+ . η− − + − + < −A1 < η− < ξ < 0 < −ξ < η+ < A1 < η+ . η−
10.3.2. On trajectories going to infinity on phase plane. Under condition (15), for any h ∈ R, a system of the form (92) has no phase trajectory having the horizontal or vertical asymptote on the phase plane in a neighborhood of an infinitely distant point (to prove this, we use the methods of Sec. 4.6). Lemma 10.1. Under condition (15), for any h = 0, system (92) (ω = Ω) has trajectories going to infinity and tending to certain lines of the family defined by the linear differential form: dΩ + hdα = 0. Moreover, in any small neighborhood of any such trajectory, other trajectories tend to it. On the phase cylinder, there exists a unique (with accuracy up to central symmetry) line that is an asymptote for trajectories going to infinity. Lemma 10.2. An infinitely distant point is an attracting (for h > 0) and repelling (for h < 0) complicated node. Corollary 10.1. For h < 0 (for h > 0), “most” phase trajectories of system (92) on the plane have an infinitely distant point as the ω-limit (α-limit) set. 877
Table 1 Number of Range of Equilibrium point type Number of subcases change of h (0, 0) (π, 0) picture 1 (−∞, −ξ) UN S 31 − 2 (−ξ, η− ) UN UN 37∗ − − 3 (η− , η+ ) UF UN 37∗ − 4 (η+ , −A1 ) UF UF 37 5 (−A1 , 0) UF SF UC-33 6 (0, A1 ) UF SF SC-32 + 7 (A1 , η− ) SF SF 36 + + 8 (η− , η+ ) SN SF 36∗ + 9 (η+ , ξ) SN SN 36∗ 10 (ξ, +∞) S SN 30
Table 2 Number of Range of Equilibrium point type Number of subcases change of h (0, 0) (π, 0) picture − 1 (−∞, η− ) UN S 31 − 2 (η− , −ξ) UF S 31∗ − 3 (−ξ, η+ ) UF UN 37∗ − 4 (η+ , −A1 ) UF UF 37 5 (−A1 , 0) UF SF UC-33 6 (0, A1 ) UF SF SC-32 + 7 (A1 , η− ) SF SF 36 + 8 (η− , ξ) SN SF 36∗ + 9 (ξ, η+ ) S SF 30∗ + 10 (η+ , +∞) S SN 30
10.3.3. On closed trajectories of different topological types on the phase cylinder. Let us consider the problem of existence of closed trajectories of different topological types on the phase cylinder for systems of the form (92). First we study the problem of existence of trajectories not contractible to a point along the phase cylinder, i.e., the trajectories enveloping the phase cylinder. Lemma 10.3. Under condition (15), for h = 0, systems of the form (92) have no closed phase trajectory contractible to a point along the phase cylinder. Proof. To system (92), we put in correspondence the system α = u,
d F(α) F(α) − h u − A1 , u = A2 F(α) + A1 u dα cos α cos α
878
(223)
Table 3 Number of Range of Equilibrium point type Number of subcase change of h (0, 0) (π, 0) picture − 1 (−∞, η− ) UN S 31 − 2 (η− , −A1 ) UF S 31∗ 3 (−A1 , −ξ) UF S 31∗ − 4 (−ξ, η+ ) UF SN 35∗ − 5 (η+ , 0) UF SF 35 + 6 (0, η− ) UF SF 34 + 7 (η− , ξ) UN SF 34∗ 8 (ξ, A1 ) S SF 30∗ + 9 (A1 , η+ ) S SF 30∗ + 10 (η+ , +∞) S SN 30 Table 4 Number of Range of Equilibrium point type Number of subcase change of h (0, 0) (π, 0) picture − 1 (−∞, η− ) UN S 31 − 2 (η− , −A1 ) UF S 31∗ + 3 (−A1 , η− ) UF S 31∗ + 4 (η− , −ξ) UN S 31 5 (−ξ, 0) UN SN 35∗ 6 (0, ξ) UN SN 34 − 7 (ξ, η+ ) S SN 30 − 8 (η+ , A1 ) S SF 30∗ + 9 (A1 , η+ ) S SF 30∗ + 10 (η+ , +∞) S SN 30 which is obtained by the nonsingular transformation defined as F(α) u = Ω + A1 cos α and which transforms the desired phase trajectories of system (92) to analogous trajectories of system (223). System (223) for h = 0 is denoted by (223) . The characteristic function for systems (223) and (223) has the form F(α) hu u − A1 . (224) cos α It vanishes in the upper half-plane {(α, u) ∈ R2 : u > 0} only on the curve
F(α) 2 (α, u) ∈ R : u = A1 . cos α 879
The phase portrait of system 223 is depicted πin Fig. 39. System (223) has a trajectory enveloping the phase cylinder and passing through the point − 2 , ε for any ε > 0. Also, there exists the key separatrix (see Sec. 6) emanating from the point − π2 , 0 and entering the point 3 π2 , 0 . Let us prove that the key separatrices bound a domain (see Fig. 39) containing the contact curve of the vector fields of systems (223) and (223) . Indeed, the key separatrix intersects the line {(α, u) ∈ R2 : α = 0} in the upper half-plane at the point (0, u∗ ), where 0 d F (α) A2 F(α) + A1 U(α) dα ∗ cos α dα, (225) u = U(α) −π/2
and u = U(α) is the equation of the key separatrix given in the form of the graph in the upper half-plane. The right-hand side of (225) is strictly greater than W1 , where 0 W1 = A1 −π/2
The following inequality holds: α −pi/2
π d F(α) dα = −A1 F . dα cos α 2
π A2 F(a) da ≥ 0 for all α ∈ 0, , U(a) 2
u = u1 (α) is the since U(α) ≥ u1 (α), equation of the separatrix of the system (223) for A1 = 0 π where π emanating from − 2 , 0 and entering 2 , 0 . Then α π A2 F(a) F(α) F(α) π −A1 F + da + A1 > A1 for all α ∈ 0, , 2 U(a) cos α cos α 2 −π/2
and hence, for such α, α U(α) = −π/2
d F (a) A2 F(a) + A1 U(a) dα F(α) cos a da > A1 . U(a) cos α
In the strip Π( π2 ,3 π2 ) , the contact curve lies only in the closure of the lower half-plane, which proves the lemma since the desired closed trajectory enveloping the phase cylinder must lie higher than the key separatrix in the upper half-plane. The remaining part of this subsection is devoted to the study of trajectories contractible to a point along the cylinder, in particular, to the study of possible limit cycles on the plane. Note that by Lemmas 4.1 and 4.2, under certain conditions, systems of the form (92) have unique limit cycles either in the strip Π or in Π . Moreover, condition 3) of Lemma 4.1 is extra since by Lemma 4.6, the contact curves of systems (92) and (92) have real branches lying in the strip Π for h ≤ 0 and in the strip Π for h ≥ 0. According to Lemma 4.1, the space Φ falls at least into two parts (each of which is of positive measure) controlling the birth of a cycle of different stability properties. The first part CH1 = {F ∈ Φ : F(α) < F (0)|α| cos α, α ∈ (−ε, ε), α = 0} corresponds to the birth of a stable cycle to the left of the value A1 F (0), and the second part CH2 = {F ∈ Φ : F(α) > F (0)|α| cos α, α ∈ (−ε, ε), α = 0} 880
corresponds to the birth of an unstable cycle to the right of A1 F (0) (ε is sufficiently small). For example, let us consider the subclass CH1 of functions from the class Φ. Clearly, F0 (α) = AB sin α cos α ∈ CH1 (A, B > 0). Lemma 10.4. If F ∈ CH1, then in a sufficiently small neighborhood of the origin, for h > A1 F (0), systems of the form (92) have no closed phase characteristic of system (92) on the phase plane. (An analogous lemma holds for system (92) under the condition h < A1 F (0) when F ∈ CH2.) Corollary 10.2. If F (0) is the global maximum of the function d F(α) dα cos α π π for α ∈ − 2 , 2 , then on the phase plane, there are no closed curves composed of trajectories of systems of the form (92) for any h > A1 F (0). Corollary 10.3. Under the conditions of the previous corollary, we can assert that there exists h > 0 such that a unique stable limit cycle of systems of the form (92) exists in the strip Π if and only if h ∈ h, A1 F (0) . Corollary 10.4. By the latter corollary and also by a certain extended mirror symmetry, there exists h < 0 such that a unique unstable limit cycle of systems of the form (92) exists in the strip Π if and only ¯ if h ∈ (−A1 F (0), h). 10.3.4. Topological classification of system phase portraits for a certain subset of admissible functions. To complete the classification of typical phase portraits of system (220), it is necessary to study the behavior of separatrices of the existing hyperbolic saddles. For brevity, we omit this study and present a list of final results on the structure of the phase cylinder of system (220). For the cases 1.1, 1.10, 2.1, 2.2, 2.9, 2.10, 3.1, 3.2, 3.3, 3.8, 3.9, 3.10, 4.1, 4.2, 4.3, 4.4, 4.7, 4.8, 4.9, and 4.10 only for them, there are no equilibrium states (221) and (222) [263]. The information about the correspondence of subcases for four cases 1, 2, 3, and 4 to the phase portraits is shown in Tables 1–4. By star we label the portraits that differ from each other only in the topological type of certain equilibrium states (for example, instead of the node, there is a focus of the same stability and vice versa) but are (absolutely) rough. The carried-out classification of rough phase portraits for a certain subdomain of the parameter space allows us to make a conclusion about the variety of different nonequivalent topological types of phase portraits for system (220) for h2 = 0. After performing the study of the structure of the phase cylinder for system (220), in order to obtain a concrete body motion on the plane, an additional integration of the kinematic relations is necessary. 10.4. Comparisons of Some Classes of Body Motions in a Medium in the Absence and Presence of Linear Damping Moment. Let us trace the change of the global topological structure of the phase portraits for nonlinear systems describing the classes of plane-parallel rigid body motions in a medium under an additional medium action, the linear damping. First we consider the problem of the body motion under the condition of velocity constancy of the plate center of mass and the absence of the medium damping (see Sec. 6). The location of trajectories on the quasi-velocity plane (α, Ω) for this case is described by system (72) and is shown in Fig. 3. 10.4.1. General characteristic of body motion trajectories on the plane in the absence of medium damping. In Sec. 6, we have carried out the qualitative integration of the kinematic relations for finding the body location on the plane. Let us briefly list some specific location features of trajectories of system (72) on the phase cylinder {(α, Ω) ∈ R2 : α mod 2π} corresponding to some specific classes of body motion on the plane. (1) Stationary motions corresponding to equilibrium points. Along with rectilinear translational body motions perpendicular to the plane plate, there exist stationary rotations at constant angular velocity 881
(and angle of attack equal to π/2) around a point W lying on the line passing through the center of mass and the plate center (in particular, this point can be infinitely distant). The existence of such stationary motions meets principal contradictions of the following character: the opposite borders of the plate “cut” the medium in different directions. Moreover, there exist the medium resistance and its moment. The cause of such a contradiction is that for now, in the nonlinear dynamical model considered, we do not take account of the influence of rotational derivatives of the moment in the angular velocity. (2) Closed trajectories on the phase cylinder. On the phase cylinder {(α, Ω) ∈ R2 : α mod 2π}, system (72) has no closed curves contractible to a point along the cylinder, in particular, there are no limit cycles on the plane R2 {α, Ω}. System (72) has a continuum of closed phase characteristics not contractible to a point along the cylinder, i.e., enveloping the phase cylinder. (3) Existence of oscillatory and rotational domains. By the previous item (see also 6), the phase cylinder of system (72) is the closure of two domains in each of which the trajectories have their own limit sets. In the oscillatory domain (see Fig. 3), almost all trajectories “emanate” from repelling points and “enter” attracting points. The oscillatory domain is of finite area on the phase cylinder. The rotational domain is entirely filled with periodic trajectories that are their limit set and is of infinite area on the phase cylinder. The property of boundedness of plane body motions corresponding to the oscillatory and rotational domains on the quasi-velocity phase cylinder is converse to the boundedness property of these domains themselves (see Sec. 6). Indeed, unbounded plane body motions correspond to compact oscillatory domain, and the motions filling compact annulus-like domains correspond to the unbounded rotational domain. Now let us trace the change of the structure of phase trajectory location under the presence of a linear damping. 10.4.2. Characteristic of certain body trajectories on the plane in the presence of linear damping. Let us list the main specific location features of trajectories of system (220) on the phase cylinder {(α, Ω) ∈ R2 : α mod 2π} for the case h2 > 0, which, as was mentioned above, is chosen in accordance with the experiment (see Secs. 1 and 3). (1) Stationary body motions with finite angle of attack less than π. Together with rectilinear translational motions at constant velocity that exist for h2 = 0, under certain conditions, system (220) has saddles lying in the strip Π (see Figs. 32, 34, and 36). They correspond to the stationary motions of the following form: the body plate center moves along a circle centered at a fixed point of the plane; in this case, the body itself moves with constant angular velocity and constant angle of attack less than π/2 (see sec. 10.3). Note that for h2 = 0, the principal contradiction related to an analogous stationary motion for h2 = 0 described in the previous subsection is removed here. (2) Closed trajectories on the phase cylinder. For h2 = 0, on the phase cylinder, system (220) has no closed characteristics enveloping the phase cylinder, which corresponds to the principal absence of long-periodic and Poisson stable plane body trajectories in bounded domains of the plane R2 {x, y}. As is seen from Sec. 4, for h2 > 0, in the strip Π, under certain conditions, system (220) can have stable limit cycles corresponding to the stable auto-oscillations of the following form; the body executes plane angular oscillations bounded in amplitude around a certain asymptote going to infinity. In this case, the angle of attack lies on the interval 0, π2 . It is interesting to trace the formal asymptotics of the stable limit cycle size in the parameter h2 . For a finite h2 = A1 , there arises a stable limit cycle, when h2 decreases from A1 down to a certain positive value h2 , the cycle size increases as |A1 − h2 |, and for h2 = h2 , it disappears. (3) Attraction domains on the phase cylinder. For h2 = 0, system (220) becomes a nonzero mean dissipative system. As a consequence, its phase cylinder loses any distribution into the oscillatory and rotational domains. Precisely, almost the whole phase cylinder is attracted by stable singular points and 882
limit cycles. Indeed, in the case shown in Figs. 34 and 35, it is seen that the stable singular point (π, 0) attracts almost the whole phase cylinder. In Fig. 32, the point (π, 0) and the stable limit cycle also attract almost the whole cylinder, and in the case shown in Fig. 36, almost the whole phase cylinder is the attraction domain for two stable points (0, 0) and (π, 0). As for the character of the rigid body plane motions “as a whole” for h2 > 0, the main specific feature of almost all trajectories is as follows: for almost all initial conditions, the rigid body goes to infinity in contrast to the classes of plane body motions considered without taking account of the medium damping. This section is the initial stage of an applied study of the body motion in a medium taking account of the medium damping moment. Such a moment brings an additional dissipation to the system; as a result, the rectilinear translational body drag can be stable in principle. For homogeneous circular cylinders, the rectilinear translational drag is unstable for any dynamical and geometric parameters of such cylinders. Probably, it is related to the cylinder motion precisely in the water, when the medium damping is not significant, which does not allow us to speak of the stability of the rectilinear translational drag. But for cylinders having a hole inside them, under certain conditions, the above stability can be attained. Under certain conditions, account of the medium damping action on the rigid body leads to an affirmative answer to the principal question of nonlinear analysis: under the body motion in the medium with finite angles of attack, the appearance of stable auto-oscillations is possible in principle. Conclusion As was said in Sec. 1, the results of the presented work appeared owing to the study of the applied problem of the rigid body motion in a resisting medium, where we have obtained complete lists of transcendental first integrals expressed through a finite combination of elementary functions. This circumstance allows the author to carry out the complete analysis of all phase trajectories and show those of their properties that have roughness and are preserved for systems of a more general form. The complete integrability of such systems is related to their symmetries of latent type. Therefore, it is of interest to study a sufficiently wide class of dynamical systems having analogous latent symmetries. So, for example, the instability of the simplest body motion, the rectilinear translational drag, is used for methodological purposes, precisely for finding the unknown parameters of the medium action on a rigid body under quasi-stationarity conditions. The experiment in the motion of homogeneous circular cylinders in the water carried out at the Institute of Mechanics of M. V. Lomonosov State University prove that in modelling the medium action on the rigid body, it is also necessary to take account of an additional parameter that brings a dissipation to the system. In studying the class of body drags with finite angle of attack, the principal problem is finding those conditions under which there exist auto-oscillations in a finite neighborhood of the rectilinear translational drag. Therefore, there arises the necessity of a complete nonlinear study. The initial stage of such a study is neglect of the medium damping action on the rigid body. Functionally, this means the assumption that the pair of dynamical functions determining the medium action depends only on one parameter, the angle of attack. The dynamical systems arising under such a nonlinear description are variable dissipation systems. Therefore, there arises the necessity to create the methodology for studying such systems. Generally speaking, the dynamics of a rigid body interacting with a medium is just the field where there arise either nonzero mean variable dissipation systems or systems in which the energy loss in the mean during a period can vanish. In this work, we have obtained such a methodology owing to which it becomes possible to finally and analytically study a number of plane and spatial model problems. In the qualitative description of the body interaction with the medium, because of the use of the experimental information on the properties of the streamline flow-around, there arises a definite dispersion in modelling the force-model characteristics. This makes it natural to introduce the definitions of relative 883
roughness (relative structural stability) and to prove such a roughness for the systems studied. Moreover, many systems considered are merely (absolutely) Andronov–Pontryagin rough. The main results of this work are as follows: • we have presented a relatively simple methodology for finding dimensionless parameters of the medium action on a rigid body under the quasi-stationarity conditions. This methodology was successfully applied for studying the motion of bodies having a simple form, the circular cylinders entering the water; • we have elaborated new qualitative methods for studying variable dissipation systems. We have obtained the conditions for the existence of the bifurcation of stable and unstable auto-oscillation birth. We have presented the conditions for the absence of such trajectories. The theory of Poincar´e plane topographical systems and comparison systems is extended to the spatial case. We have presented a sufficiently simple methodology for proving the Poisson stability of unclosed trajectories of dynamical systems. We have introduced the definitions of relative structural stability (relative roughness) and relative structural instability (relative nonroughness) of various degrees; • we have discovered new integrable cases and families of phase portraits in the plane rigid body dynamics. We have qualitatively studied and have integrated some model variants of the body motion in a resisting medium. We have shown the first integrals of the corresponding systems that are transcendental functions and that are expressed through elementary functions. We have also studied the problem of free drag. We have constructed new two-parameter families of topologically nonequivalent phase portraits. Almost every portrait of the family is (absolutely) structurally stable; • we have discovered new integrable cases and families of many-dimensional phase portraits in the spatial rigid body dynamics. We have integrated the problem of spatial motion of a dynamically symmetric clamped rigid body placed in an over-running medium flow. We have obtained a new two-parameter family of phase portraits in the problem of the spatial free drag of a body. Almost every portrait of the family is (absolutely) Andronov–Pontryagin rough. The whole spectrum of results found under the simplest assumption of the absence of the medium damping action on a rigid body allows the author to make the conclusion that it is impossible to find those conditions under which there exist auto-oscillations in a finite neighborhood of the rectilinear translational drag. The concluding part of the work opens the first stage of studying the body motion in a medium taking the medium damping moment into account. Such a moment brings an additional dissipation to the system, and, as a result, the body rectilinear translational drag can be stable in principle. Therefore, for homogeneous circular cylinders, the rectilinear translational drag is unstable for any dynamical and geometric parameters, which is probably related to the cylinder motion precisely in the water. But for cylinders having a hole inside them, under certain conditions, it is possible to attain the stability of the rectilinear translational drag in principle. Therefore, under certain condition, the account of the medium damping action on the rigid body leads to an affirmative answer to the principal question of the nonlinear analysis: under the body motion in a medium with finite angles of attack, in principle, there can arise stable auto-oscillations. This work opens a new field of research in nonlinear analysis of the body motion in a resisting medium under quasi-stationarity conditions and taking the medium damping into account. All that was said above allows one to view the results of this work in totality as a new direction in the qualitative theory of ordinary differential equations and the dynamics of a rigid body interacting with a medium. This work was partially supported by the Russian Foundation for Basic Research (project No. 08-01-00231-a). 884
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230. V. A. Privalov and V. A. Samsonov, “On stability of motion of a body auto-rotating in medium flow,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 2, 32–38 (1990). 231. Rayleigh (J. W. Strutt), “The soaring of birds,” Nature, 28. 232. R. Reissing, G. Sansone, and R. Conti, Qualitative Theory of Ordinary Differential Equations [Russian translation], Nauka, Moscow (1974). 233. V. E. Ryzhova and M. V. Shamolin, “On some analogies in problem of body motion in a resisting medium,” in: Seventh Congress in Theoretical and Applied Mechanics, Moscow, August 15–21, 1991 [in Russian], Moscow (1991). 234. S. T. Sadetov, “Integrability conditions of Kirchhoff equations,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 56–62 (1990). 235. T. V. Sal’nikova, “On integrability of Kirchhoff equations in symmetric case,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 4, 68–71 (1985). 236. V. A. Samsonov, “On stability of solutions of systems of differential equations in some cases,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 74–78 (1962). 237. V. A. Samsonov, “On stability of equilibrium of physical pendulum with fluid filling,” Prikl. Mat. Mekh., 30, No. 6, 1112–1114 (1966). 238. V. A. Samsonov, “On problem of minimum of a functional in studying stability of motion of a body with fluid filling,” Prikl. Mat. Mekh., 31, No. 3, 523–526 (1967). 239. V. A. Samsonov, “On quasi-stationary motions of mechanical systems,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 1, 32–50 (1978). 240. V. A. Samsonov, Issues on Mechanics. Some Problems, Phenomena, and Paradoxes [in Russian], Nauka, Moscow (1980). 241. V. A. Samsonov, “On body rotation in magnetic field,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 4, 32–34 (1984). 242. V. A. Samsonov, V. A. Eroshin, G. A. Konstantinov, and V. M. Makarshin, “Two model problems on body motion in a medium under streamline flow-around,” in: Scientific Report of Institute of Mechanics, Moscow State Univ., No. 3427 [in Russian], Institute of Mechanics, Moscow State Univ., Moscow (1987). 243. V. A. Samsonov, V. V. Kozlov, and M. V. Shamolin, “Reduction in problem of body motion in a medium under steamline flow-around,” in: Modern Problems of Mechanics and Technologies of Mashine Industry, All-Union Conf. (April 16–18, 1989). A Plenary Report [in Russian], All-Union Institute for Scientific and Technical Information, Moscow (1989). 244. V. A. Samsonov, B. Ya. Lokshin, and V. A. Privalov, “Qualitative analysis,” in: Scientific Report of Institute of Mechanics, Moscow State Univ., No. 3245 [in Russian], Institute of Mechanics, Moscow State Univ., Moscow (1985). 245. V. A. Samsonov and M. V. Shamolin, “On body motion in a resisting medium,” in: Contemporary Problems of Mechanics and Technologies of Machine Industry, All-Union Conf., April 16–18, 1989. Abstracts of Reports [in Russian], All-Union Institute for Scientific and Technical Information, Moscow (1989), pp. 128–129. 246. V. A. Samsonov and M. V. Shamolin, “To problem on body motion in a resisting medium,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 51–54 (1989). 247. V. A. Samsonov and M. V. Shamolin, “A model problem of body motion in a medium with streamline flow-around,” Scientific Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 3969, Institute of Mechanics, Moscow State Univ., Moscow (1990). 248. V. A. Samsonov and M. V. Shamolin, “A model problem of body motion in a medium with streamline flow-around,” in: Nonlinear Oscillations of Mechanical Systems, Abstracts of Reports of II All-Union Conf., September, 1990, Pt. 2 [in Russian], Gor’kii (1990), pp. 95–96. 895
249. V. A. Samsonov and M. V. Shamolin, “To problem of body drag in a medium under streamline flow-around,” in: Scientific Report of Institute of Mechanics, Moscow State Univ., No. 4141 [in Russian], Institute of Mechanics, Moscow State Univ., Moscow (1991). 250. V. A. Samsonov and M. V. Shamolin, “On stability of a body under its drag in a resisting medium,” in: VII Chetaev Conf. “Analytical Mechanics, Stability, and Control of Motion,” June 10–13, 1997. Abstracts of Reports, Kazan’ State Technical Univ., Kazan’ (1997), p. 24. 251. V. A. Samsonov, M. V. Shamolin, V. A. Eroshin, and V. M. Makarshin, “Mathematical modelling in problem of body drag in a resisting medium under streamline flow-around,” in: Scientific Report of Institute of Mechanics, Moscow State Univ., No. 4396 [in Russian], Moscow (1995). 252. G. Sansone, Ordinary Differential Equations [Russian translation], Inostr. Lit., Moscow (1954). 253. L. I. Sedov, Mechanics of Continuous Media [in Russian], Vols., 1, 2, Nauka, Moscow (1983, 1984). 254. H. Seifert and W. Threifall, Topology [Russian translation], Gostekhizdat, Moscow (1938). 255. N. Yu. Selivanova and M. V. Shamolin, “Extended Cahn–Hilliard model and certain of its solutions,” in: Materials of Voronezh All-Russian Conf. “Pontryagin Readings—XVIII,” Voronezh, May, 2007, Voronezh State Univ., Voronezh (2007), pp. 145–146. 256. J. L. Singh, Classical Dynamics [Russian translation], Fizmatgiz, Moscow (1963). 257. M. V. Shamolin, Qualitative Analysis of a Model Problem of Body Motion in a Medium with Streamline Flow-Around [in Russian], Candidate Dissertation, MGU, Moscow (1991). 258. M. V. Shamolin, Qualitative Analysis of a Model Problem of Body Motion in a Medium with Streamline Flow-Around [in Russian], Theses of Candidate Dissertation, MGU, Moscow (1991). 259. M. V. Shamolin, “Closed trajectories of different topological types in problem of body motion in a medium with resistance,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 52–56 (1992). 260. M. V. Shamolin, “To problem of body motion in a medium with resistance,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 52–58 (1992). 261. M. V. Shamolin, “A new two-parameter family of phase portaits for problem of a body motion in a resisting medium,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 24–28, 1993. Abstracts of Reports, Pt. 2 [in Russian], Znanie, Kiev (1993), pp. 62–63. 262. M. V. Shamolin, “Applications of Poincar´e topographical system methods and comparison systems in some concrete systems of differential equations,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 66–70 (1993). 263. M. V. Shamolin, “Classification of phase portraits in problem of body motion in a resisting medium in the presence of a linear damping moment,” Prikl. Mat. Mekh., 57, No. 4, 40–49 (1993). 264. M. V. Shamolin, “Existence and uniqueness of trajectories having infinitely distant points as limit sets for dynamical systems on a plane,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 68–71 (1993). 265. M. V. Shamolin, “Global qualitative analysis of nonlinear systems on the problem of a body motion in a resisting medium,” in: Fourth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, August 18–21, 1993, Szeged, Hungary (1993), p. 54. 266. M. V. Shamolin, “Relative structural stability of problem of body motion in a resisting medium,” in: Mechanics and Its Applications, Sci. Conf., November 9–11, 1993, Abstracts of Reports, Tashkent State Univ., Tashkent (1993), pp. 20–21. 267. M. V. Shamolin, “A new two-parameter family of phase portraits in problem of a body motion in a medium,” Dokl. Ross. Akad. Nauk, 337, No. 5, 611–614 (1994). 268. M. V. Shamolin, “On relative roughness of dynamical systems in problem of body motion in a medium under streamline flow around,” in: Modelling and Study Stability of Systems, Sci. Conf., May 16–20, 1994. Abstract of Reports [in Russian], Kiev (1994), pp. 144–145. 269. M. V. Shamolin, “Relative structural stability on the problem of a body motion in a resisting medium,” in: ICM’94, Abstract of Short Communications, Zurich, 3–11 August, 1994, Zurich, Switzerland (1994), p. 207. 896
270. M. V. Shamolin, “A new two-parameter family of phase portraits with limit cycles in rigid body dynamics interacting with a medium,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 15–19, 1995. Abstracts of Reports (Study of Systems) [in Russian], Kiev (1995), p. 125. 271. M. V. Shamolin, “New two-parameter families of the phase patterns on the problem of a body motion in a resisting medium,” in: ICIAM’95, Book of Abstracts. Hamburg, 3–7 July, 1995, Hamburg, Germany (1995), p. 436. 272. M. V. Shamolin, “On relative structural stability of dynamical systems in problem of body motion in a resisting medium,” the abstract of a talk at the Chebyshev Readings, Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 6, 17 (1995). 273. M. V. Shamolin, “Poisson-stable and dense orbits in rigid body dynamics,” in: 3rd Experimental Chaos Conf., Advance Program. Edinburgh, Scotland, August 21–23, 1995, Edinburgh, Scotland (1995), p. 114. 274. M. V. Shamolin, “Qualitative methods to the dynamic model of interaction of a rigid body with a resisting medium and new two-parametric families of the phase portraits,” in: DynDays’95 (Sixteenth Annual Informal Workshop), Program and Abstracts. Lyon, June 28–July 1, 1995, Lyon, France (1995), p. 185. 275. M. V. Shamolin, “Relative structural stability of dynamical systems for problem of body motion in a medium,” in: Analytical, Numerical, and Experimental Methods in Mechanics. A Collection of Scientific Works [in Russian], Izd. Mosk. Univ., Moscow (1995), pp. 14–19. 276. M. V. Shamolin, “Structural optimization of the controlled rigid motion in a resisting medium,” in: WCSMO-1, Extended Abstracts. Posters. Goslar, May 28–June 2, 1995, Goslar, Germany (1995), p. 18–19. 277. M. V. Shamolin, “A list of integrals of dynamical equations in spatial problem of body motion in a resisting medium,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 20–24, 1996. Abstracts of Reports (Study of Systems) [in Russian], Kiev (1996), p. 142. 278. M. V. Shamolin, “Definition of relative roughness and two-parameter family of phase portraits in rigid body dynamics,” Usp. Mat. Nauk, 51, No. 1, 175–176 (1996). 279. M. V. Shamolin, “Introduction to problem of body drag in a resisting medium and a new two-parameter family of phase portraits,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 4, 57–69 (1996). 280. M. V. Shamolin, “Introduction to spatial dynamics of rigid body motion in resisting medium,” in: Materials of Int. Conf. and Chebyshev Readings Devoted to the 175th Anniversary of P. L. Chebyshev, Moscow, May 14–19, 1996, Vol. 2 [in Russian], Izd. Mosk. Univ., Moscow (1996), pp. 371–373. 281. M. V. Shamolin, “On a certain integrable case in dynamics of spatial body motion in a resisting medium,” in: II Symposium in Classical and Celestial Mechanics. Abstracts of Reports. Velikie Luki, August 23–28, 1996 [in Russian], Moscow–Velikie Luki (1996), pp. 91–92. 282. M. V. Shamolin, “Periodic and Poisson stable trajectories in problem of body motion in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 55–63 (1996). 283. M. V. Shamolin, “Qualitative methods in dynamics of a rigid body interacting with a medium,” in: II Siberian Congress in Applied and Industrial Mathematics, Novosibirsk, June 25–30, 1996. Abstracts of Reports, Pt. III [in Russian], Novosibirsk (1996), p. 267. 284. M. V. Shamolin, “Qualitative methods in interacting with the medium rigid body dynamics,” in: Abstracts of GAMM Wissenschaftliche Jahrestagung’96, 27.–31. May, 1996, Czech Rep., Karls-Universit¨ at Prag., Prague (1996), pp. 129–130. 285. M. V. Shamolin, “Qualitative methods in interacting with the medium rigid body dynamics,” in: Abstracts of XIXth ICTAM, Kyoto, Japan, August 25–31, 1996, Kyoto, Japan (1996), p. 285. 286. M. V. Shamolin, “Relative structural stability and relative structural instability of different degrees in topological dynamics,” in: Abstracts of Int. Topological Conf. Dedicated to P. S. Alexandroff ’s 897
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100th Birthday “Topology and Applications,” Moscow, May 27–31, 1996, Fazis, Moscow (1996), pp. 207–208. M. V. Shamolin, “Spatial Poincar´e topographical systems and comparison systems,” in: Abstract of Reports of Math. Conf. “Erugin Readings,” Brest, May 14–16, 1996 [in Russian], Brest (1996), p. 107. M. V. Shamolin, “Topographical Poincar´e systems in many-dimensional spaces,” in: Fifth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, Regional Committee of the Hungarian Academy of Sciences, July 29–August 2, 1996, Szeged, Hungary (1996), p. 45. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium,” Dokl. Ross. Akad. Nauk, 349, No. 2, 193–197 (1996). M. V. Shamolin, “Classical problem of a three-dimensional motion of a pendulum in a jet flow,” in: 3rd EUROMECH Solid Mechanics Conf., Book of Abstracts, Stockholm, Sweden, August 18–22, 1997, Royal Inst. of Technology, Stockholm, Sweden (1997), p. 204. M. V. Shamolin, “Families of three-dimensional phase portraits in dynamics of a rigid body,” in: EQUADIFF 9, Abstracts, Enlarged Abstracts, Brno, Czech Rep., August 25–29, 1997, Masaryk Univ., Brno, Czech Rep. (1997), p. 76. M. V. Shamolin, “Jacobi integrability of problem of a spatial pendulum placed in over-running medium flow,” in: Modelling and Investigation of System Stability. Sci. Conf., May 19–23, 1997. Abstracts of Reports [in Russian], Kiev (1997), p. 143. M. V. Shamolin, “Mathematical modelling of dynamics of a spatial pendulum flowing around by a medium,” in Proc. of VII Int. Symposium “Methods of Discrete Singularities in Problems of Mathematical Physics,” Feodociya, June 26–29, 1997 [in Russian], Kherson State Technical Univ., Kherson (1997), pp. 153–154. M. V. Shamolin, “On an integrable case in spatial dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 65–68 (1997). M. V. Shamolin, “Partial stabilization of body rotational motions in a medium under a free drag,” Abstracts of Reports of All-Russian Conf. with Int. Participation “Problems of Celestial Mechanics,” St. Petersburg, June 3–6, 1997 [in Russian], Institute of Theoretical Astronomy, Russian Academy of Sciences, St. Petersburg (1997), pp. 183–184. M. V. Shamolin, “Qualitative methods in dynamics of a rigid body interacting with a medium,” in: YSTM’96: “Young People, the Third Millenium,” Proc. of Int. Congress (Ser. Professional ) [in Russian], Vol. 2, NTA “APFN,” Moscow (1997), pp. I-4. M. V. Shamolin, “Spatial dynamics of a rigid body interacting with a medium,” Workshop in Mechanics of Systems and Problems of Motion Control and Navigation, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 4, 174 (1997). M. V. Shamolin, “Spatial Poincar´e topographical systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997). M. V. Shamolin, “Three-dimensional structural optimization of controlled rigid motion in a resisting medium,” in: Proc. of WCSMO-2, Zakopane, Poland, May 26–30, 1997, Zakopane, Poland (1997), pp. 387–392. M. V. Shamolin, “Three-dimensional structural optimization of controlled rigid motion in a resisting medium,” in: WCSMO-2, Extended Abstracts, Zakopane, Poland, May 26–30, 1997, Zakopane, Poland (1997), pp. 276–277. M. V. Shamolin, “Absolute and relative structural stability in spatial dynamics of a rigid body interacting with a medium,” in: Proc. of Int. Conf. “Mathematics in Industry,” ICIM-98, Taganrog, June 29–July 3, 1998 [in Russian], Taganrog State Pedagogical Inst., Taganrog (1998), pp. 332–333. M. V. Shamolin, “Families of portraits with limit cycles in plane dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 29–37 (1998).
303. M. V. Shamolin, “Family of three-dimensional phase portraits in spatial dynamics of a rigid body interacting with a medium,” in: III Int. Symposium in Classical and Celestial Mechanics, August 23–27, 1998, Velikie Luki. Abstracts of Reports [in Russian], Computational Center of Russian Academy of Sciences, Moscow–Velikie Luki (1998), pp. 165–167. 304. M. V. Shamolin, “Lyapunov functions method and many-dimensional topographical systems of Poincar´e in rigid body dynamics,” in: Abstract of Reports of IV Crimenian Int. Math. School “Lyapunov Function Method and Its Applications,” Crimea, Alushta, September 5–12, Simpheropol’ State Univ., Simpheropol’ (1998), p. 80. 305. M. V. Shamolin, “Many-dimensional topographical Poincar´e systems in rigid body dynamics,” in: Abstracts of GAMM Wissenschaftliche Jahrestagung’98, 6.–9. April, 1998, Universit¨ at Bremen, Bremen, Germany (1998), p. 128. 306. M. V. Shamolin, “Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of Int. Congress “Nonlinear Analysis and Its Applications,” Moscow, September 1–5, 1998 [in Russian], Moscow (1998), p. 131. 307. M. V. Shamolin, “Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” in: CD-Proc. of the Congress “Nonlinear Analysis and Its Applications,” Moscow, Russia, September 1–5, 1998, Moscow (1999), pp. 497–508. 308. M. V. Shamolin, “New two-parametric families of the phase portraits in three-dimensional rigid body dynamics,” in: Int. Conf. Devoted to the 90th Anniversary of L. S. Pontryagin, Moscow, August 31–September 6, 1998, Abstract of Reports, Differntial Equations, Izd. Mosk. Univ., Moscow (1998), pp. 97–99. 309. M. V. Shamolin, “On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998). 310. M. V. Shamolin, “Qualitative and numerical methods in some problems of spatial dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of 5th Int. Conf.-Workshop “Engineering-Physical Problems of New Tehnics,” Moscow, May 19–22, 1998 [in Russian], Moscow State Technical Univ., Moscow (1998), pp. 154–155. 311. M. V. Shamolin, “Some classical problems in three-dimensional dynamics of a rigid body interacting with a medium,” in: Proc. of ICTACEM’98, Kharagpur, India, December 1–5, 1998, Aerospace Engineering Dep., Indian Inst. of Technology, Kharagpur, India (1998), p. 11. 312. M. V. Shamolin, “Some problems of spatial dynamics of a rigid body interactng with a medium under quasi-stationarity conditions,” in: Abstracts of Reports of All-Russian Sci.-Tech. Conf. of Young Scientists “Modern Problems of Aero-Cosmos Science,” Zhukovskii, May 27–29, 1998 [in Russian], Central Aero-Hydrodynamical Inst., Moscow (1998), pp. 89–90. 313. M. V. Shamolin, “Certain classes of partial solutions in dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 178–189 (1999). 314. M. V. Shamolin, “Families of long-period trajectories in spatial dynamics of a rigid body,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 25–29 1999. Abstracts of Reports [in Russian], Kiev (1999), p. 60. 315. M. V. Shamolin, “Integrability in terms of transcendental functions in rigid body dynamics,” in: Book of Abstracts of GAMM Annual Meeting, April 12–16, 1999, Metz, France, Universit´e de Metz, Metz, France (1999), p. 144. 316. M. V. Shamolin, “Long-periodic trajectories in rigid body dynamics,” in: Sixth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, Regional Committee of the Hungarian Academy of Sciences, August 10–14, 1999, Szeged, Hungary (1999), p. 47. 317. M. V. Shamolin, “Mathematical modelling in 3D dynamics of a rigid interacting with a medium,” in: Book of Abstracts of the Second Int. Conf. “Tools for Mathematical Modelling,” Saint-Petersburg, Russia, 14–19 June, 1999, Saint-Petersburg State Tech. Univ., Saint-Petersburg (1999), pp. 122–123. 899
318. M. V. Shamolin, “Methods of analysis of a deceleration of a rigid in 3D medium,” in: Contributed Abstracts of 3rd ENOC, Copenghagen (Lyngby), Denmark, August 8–12, 1999, Tech. Univ. of Denmark, Copenghagen (1999). 319. M. V. Shamolin, “New families of the nonequivalent phase portraits in 3D rigid body dynamics,” in: Abstracts of Second Congress ISAAC 1999, Fukuoka, Japan, August 16–21, 1999, Fukuoka Ins. of Tech., Fukuoka (1999), pp. 205–206. 320. M. V. Shamolin, “New Jacobi integrable cases in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999). 321. M. V. Shamolin, “Nonlinear dynamical effects in spatial body drag in a resisting medium,” in: Abstracts of Reports of III Int. Conf. “Chkalov Readings, Engineering-Physical Problems of Aviation and Cosmos Technics” (June 1–4, 1999 ) [in Russian], EATK GA, Egor’evsk (1999), pp. 257–258. 322. M. V. Shamolin, “On roughness of dissipative systems and relative roughness and nonroughness of variable dissipation systems,” Usp. Mat. Nauk, 54, No. 5, 181–182 (1999). 323. M. V. Shamolin, “Properties of integrability of systems in terms of transcendental functions,” in: Final Progr. and Abstracts of Fifth SIAM Conf. on Appl. of Dynamic. Syst., May 23–27, 1999, Snowbird, Utah, USA, SIAM (1999), p. 60. 324. M. V. Shamolin, “Some properties of transcendental integrable dynamical systems,” in: Book of Abstracts of EQUADIFF 10, Berlin, August 1–7, 1999, Free Univ. of Berlin, Berlin (1999), pp. 286–287. 325. M. V. Shamolin, “Structural stability in 3D dynamics of a rigid body,” in: WCSMO-3, Short Paper Proc., Buffalo, NY, May 17–21, 1999, Vol. 2, Buffalo (1999), pp. 475–477. 326. M. V. Shamolin, “Structural stability in 3D dynamics of a rigid body,” in: CD-Proc. of WCSMO-3, Buffalo, NY, May 17–21, 1999, Buffalo (1999), p. 6. 327. M. V. Shamolin, “A new family of phase portraits in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 371, No. 4, 480–483 (2000). 328. M. V. Shamolin, “About interaction of a rigid body with a resisting medium under an assumption of a jet flow,” in: Book of Abstracts II (General sessions) of 4th EUROMECH Solid Mech. Conf., Metz, France (June 26–30, 2000), Univ. of Metz (2000), p. 703. 329. M. V. Shamolin, “Comparison of certain integrability cases from two-, three-, and four-dimensional dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of V Crimeanian Int. Math. School “Lyapunov Function Method and Its Application,” (MLF-2000), Crimea, Alushta, September 5–13, 2000 [in Russian], Simpheropol’ (2000), p. 169. 330. M. V. Shamolin, “Integrability and nonintegrability in terms of transcendental functions,” in: CD-Abstracts of 3rd ECM (Poster sessions), Barcelona, Spain, June 10–14, 2000, Poster No. 36. 331. M. V. Shamolin, “Jacobi integrability of problem of four-dimensional body motion in a resisting medium,” in: Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, August 21–26, 2000 [in Russian], Vladimir State Univ., Vladimir (2000), pp. 196–197. 332. M. V. Shamolin, “Jacobi integrability in problem of four-dimensional rigid body motion in a resisting medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000). 333. M. V. Shamolin, “Many-dimensional Poincar´e systems and transcendental integrability,” in: IV Siberian Congress in Applied and Industrial Mathematics, Novosibirsk, June 26–July 1, 2000. Abstracts of Reports, Pt. I [in Russian], Novosibirsk, Institute of Mathematics (2000), pp. 25–26. 334. M. V. Shamolin, “Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability,” In: Book of Abstracts of ECCOMAS 2000, Barcelona, Spain, 11–14 September, Barcelona (2000), p. 495. 335. M. V. Shamolin, “Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability,” in: CD-Proc. of ECCOMAS 2000, Barcelona, Spain, 11–14 September, Barcelona (2000), p. 11. 900
336. M. V. Shamolin, “Methods of analysis of dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of Annual Sci. Conf. GAMM 2000 at the Univ. of G¨ ottingen, 2–7 April, 2000, Univ. of G¨ ottingen (2000), p. 144. 337. M. V. Shamolin, “New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of 16th IMACS World Congress 2000, Lausanne, Switzerland, August 21–25, 2000, EPFL (2000), p. 283. 338. M. V. Shamolin, “New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium,” in: CD-Proc. of 16th IMACS World Congress 2000, Lausanne, Switzerland, August 21–25, 2000, EPFL (2000). 339. M. V. Shamolin, “On a certain case of Jacobi integrability in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Int. Conf. in Differential and Integral Equations, Odessa, September 12–14, 2000 [in Russian], AstroPrint, Odessa (2000), pp. 294–295. 340. M. V. Shamolin, “On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000). 341. M. V. Shamolin, “On roughness of disspative systems and relative roughness of variable dissipation systems,” the abstract of a talk at the Workshop in Vector and Tensor Analysis Named after P. K. Rashevskii, Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 63 (2000). 342. M. V. Shamolin, “Problem of four-dimensional body motion in a resisting medium and one case of integrability,” in: Book of Abstracts of the Third Int. Conf. “Differential Equations and Applications,” St. Petersburg, Russia, June 12–17, 2000 [in Russian], St. Petersburg State Univ., St. Petersburg (2000), p. 198. 343. M. V. Shamolin, “Comparison of some cases of integrability in dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of Annual Sci. Conf. GAMM 2001, ETH Zurich, 12–15 February, 2001, ETH, Zurich (2001), p. 132. 344. M. V. Shamolin, “Complete integrability of equations for motion of a spatial pendulum in over-running medium flow,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 22–28 (2001). 345. M. V. Shamolin, “Diagnosis problem as the main problem of general differential diagnosis problem,” in: Book of Abstracts of the Third Int. Conf. “Tools for Mathematical Modelling,” St. Petersburg, Russia, June 18–23, 2001 [in Russian], St. Petersburg State Technical Univ., St. Petersburg (2001), p. 121. 346. M. V. Shamolin, “Integrability cases of equations for spatial dynamics of a rigid body,” Prikl. Mekh., 37, No. 6, 74–82 (2001). 347. M. V. Shamolin, “Integrability of a problem of four-dimensional rigid body in a resisting medium,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” Fund. Prikl. Mat., 7, No. 1, 309 (2001). 348. M. V. Shamolin, “New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Sci. Conf., May 22–25, 2001 [in Russian], Kiev (2001), p. 344. 349. M. V. Shamolin, “New Jacobi integrable cases in dynamics of two-, three-, and four-dimensional rigid body interacting with a meduium,” in: Abstracts of Reports of VIII All-Russian Congress in Theoretical and Applied Mechanics, Perm’, August 23–29, 2001 [in Russian], Ural Department of the Russian Academy of Sciences, Ekaterinburg (2001), pp. 599–600. 350. M. V. Shamolin, “On stability of motion of a body twisted around its longitudinal axis in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 189–193 (2001). 351. M. V. Shamolin, “Pattern recognition in the model of the interaction of a rigid body with a resisting medium,” in: Col. of Abstracts of First SIAM–EMS Conf. “Applied Mathematics in Our Changing World,” Berlin, Germany, September 2–6, 2001, Springer, Birkh¨ auser (2001), p. 66. 901
352. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a medium,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” Fund. Prikl. Mat., 7, No. 1, 302–303 (2001). 353. M. V. Shamolin, “Dynamical systems with variable dissipation in 3D dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of 4th ENOC, Moscow, Russia, August 19–23, 2002, Inst. Probl. Mech. Russ. Acad. Sci., Moscow (2002), p. 109. 354. M. V. Shamolin, “Foundations in differential and topological diagnostics,” in: Book of Abstracts of Annual Sci. Conf. GAMM 2002, Univ. of Augsburg, March 25–28, 2002, Univ. of Augsburg (2002), p. 154. 355. M. V. Shamolin, “Methods of analysis of dynamics of a 2D- 3D-, or 4D-rigid body with a medium,” in: Abstracts, Short Communications, Poster Sessions of ICM-2002, Beijing, August 20–28, 2002, Higher Education Press, Beijing, China (2002), p. 268. 356. M. V. Shamolin, “New integrable cases in dynamics of a two-, three-, and four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, July 1–6, 2002 [in Russian], Vladimir State Univ., Vladimir (2002), pp. 142–144. 357. M. V. Shamolin, “On integrability of certain classes of nonconservative systems,” Usp. Mat. Nauk, 57, No. 1, 169–170 (2002). 358. M. V. Shamolin, “Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium,” J. Math. Sci., 110, No. 2, 2526–2555 (2002). 359. M. V. Shamolin, “Foundations of differential and topological diagnostics,” J. Math. Sci., 114, No. 1, 976–1024 (2003). 360. M. V. Shamolin, “Global structural stability in dynamics of a rigid body interacting with a medium,” in: 5th ICIAM, Sydney, Australia, 7–11 July, 2003, Univ. of Technology, Sydney (2003), p. 306. 361. M. V. Shamolin, “Integrability and nonintegrability in terms of transcendental functions,” in: Book of Abstracts of Annual Sci. Conf. GAMM 2003, Abano Terme–Padua, Italy, 24–28 March, 2003, Univ. of Padua (2003), p. 77. 362. M. V. Shamolin, “Integrability in transcendental functions in rigid body dynamics,” in: Abstracts of Reports of Sci. Conf. “Lomonosov Readings,” Sec. Mechanics, April 17–27, 2003, Moscow, M. V. Lomonosov Moscow State Univ. [in Russian], MGU, Noscow (2003), p. 130. 363. M. V. Shamolin, “New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium,” J. Math. Sci., 114, No. 1, 919–975 (2003). 364. M. V. Shamolin, “On a certain spatial problem of rigid body motion in a resisting medium,” in: Abstracts of Reports of Int. Sci. Conf. “Third Polyakhov Readings,” St. Petersburg, February 4–6, 2003 [in Russian], NIIKh St. Petersburg Univ., St. Petersburg (2003), pp. 170–171. 365. M. V. Shamolin, “On integrability of nonconservative dynamical systems in transcendental functions,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 27–30, 2003, Abstracts of Reports [in Russian], Kiev (2003), p. 277. 366. M. V. Shamolin, “Some questions of differential and topological diagnostics,” in: Book of Abstracts of 5th European Solid Mech. Conf. (ESMC-5), Thessaloniki, Greece, August 17–22, 2003, Aristotle Univ. Thes. (AUT), European Mech. Sc. (EUROMECH) (2003), p. 301. 367. M. V. Shamolin, “Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body,” J. Math. Sci., 122, No. 1, 2841–2915 (2004). 368. M. V. Shamolin, “Geometric representation of motion in a certain problem of body interaction with a medium,” Prikl. Mekh., 40, No. 4, 137–144 (2004). 369. M. V. Shamolin, “Integrability of nonconservative systems in elementary functions,” in: X Math. Int. Conf. Named after Academician M. Kravchuk, May 13–15, 2004, Kiev [in Russian], Kiev (2004), p. 279. 902
370. M. V. Shamolin, Methods for Analysis of Classes of Nonconservative Systems in Dynamics of a Rigid Body Interacting with a Medium [in Russian], Doctoral Dissertation, MGU, Moscow (2004). 371. M. V. Shamolin, Methods for Analysis of Classes of Nonconservative Systems in Dynamics of a Rigid Body Interacting with a Medium [in Russian], Theses of Doctoral Dissertation, MGU, Moscow (2004). 372. M. V. Shamolin, “Some cases of integrability in dynamics of a rigid body interacting with a resisting medium,” in: Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, July 5–10, 2004, Vladimir State Univ., Vladimir (2004), pp. 296–298. 373. M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], Ekzamen, Moscow (2004). 374. M. V. Shamolin, “A case of complete integrability in spatial dynamics of a rigid body interacting with a medium taking account of rotational derivatives of force moment in angular velocity,” Dokl. Ross. Akad. Nauk, 403, No. 4, 482–485 (2005). 375. M. V. Shamolin, “Cases of complete integrability in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Int. Conf. “Functional Spaces, Approximation Theory, and Nonlinear Analysis” Devoted to the 100th Anniversary of A. M. Nikol’skii, Moscow, May 23–29, 2005 [in Russian], V. A. Steklov Math. Inst. of the Russian Academy of Sciences, Moscow (2005), p. 244. 376. M. V. Shamolin, “Comparison of Jacobi integrable cases of plane and spatial body motions in a medium under streamline flow-around,” Prikl. Mat. Mekh., 69, No. 6, 1003–1010 (2005). 377. M. V. Shamolin, “Integrability in transcendental functions in rigid body dynamics,” in: Math. Conf. “Modern Problems of Applied Mathematics and Mathematical Modelling,” Voronezh, December 12–17, 2005 [in Russian], Voronezh State Acad., Voronezh (2005), p. 240. 378. M. V. Shamolin, “Mathematical model of interaction of a rigid body with a resisting medium in a jet flow,” in: Abstracts. Pt. 1. 76 Annual Sci. Conf. (GAMM), Luxembourg, March 28–April 1, 2005, Univ. du Luxembourg (2005), pp. 94–95. 379. M. V. Shamolin, “On a certain integrable case in dynamics on so(4) × R4 ,” in: Abstracts of Reports of All-Russian Conf. “Differential Equations and Their Applications,” (SamDif-2005), Samara, June 27–July 2, 2005 [in Russian], Univers-Grupp, Samara (2005), pp. 97–98. 380. M. V. Shamolin, “On a certain integrable case of equations of dynamics in so(4) × R4 ,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005). 381. M. V. Shamolin, “On body motion in a resisting medium taking account of rotational derivatives of aerodynamic force moment in angular velocity,” in: Abstracts of Reports of Sci. Conf. “Lomonosov Readings-2005,” Sec. Mechanics, April, 2005, Moscow, M. V. Lomonosov Moscow State Univ. [in Russian], MGU, Moscow (2005), p. 182. 382. M. V. Shamolin, “On rigid body motion in a resisting medium taking account of rotational derivatives of areodynamical force moment in angular velocity,” in: Modelling and Studying of Systems, Sci. Conf., May 23–25, 2005. Abstracts of Reports [in Russian], Kiev (2005), p. 351. 383. M. V. Shamolin, “Some cases of integrability in 3D dynamics of a rigid body interacting with a medium,” in: Book of Abstracts. IMA Int. Conf. “Recent Advances in Nonlinear Mechanics,” Aberdeen, Scotland, August 30–September 1, 2005, Aberdeen (2005), p. 112. 384. M. V. Shamolin, “Structural stable vector fields in rigid body dynamics,” in: Abstracts of 8th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2005 ), Lodz, Poland, December 12–15, 2005, Tech. Univ. Lodz (2005), p. 78. 385. M. V. Shamolin, “Structural stable vector fields in rigid body dynamics,” in: Proc. of 8th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2005 ), Lodz, Poland, December 12–15, 2005, Vol. 1, Tech. Univ. Lodz (2005), pp. 429–436. 903
386. M. V. Shamolin, “Variable dissipation dynamical systems in dynamics of a rigid body interacting with a medium,” in: Differential Equations and Computer Algebra Tools, Materials of Int. Conf., Brest, October 5–8, 2005, Pt. 1 [in Russian], BGPU, Minsk (2005), pp. 231–233. 387. M. V. Shamolin, “Almost conservative systems in dynamics of a rigid body,” in: Book of Abstracts, 77th Annual Meeting of GAMM, March 27–31, 2006, Technische Univ. Berlin, Technische Univ. Berlin (2006), p. 74. 388. M. V. Shamolin, “Model problem of body motion in a resisting medium taking account of dependence of resistance force on angular velocity,” in: Scientifuc Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 4818, Institute of Mechanics, Moscow State Univ., Moscow (2006), p. 44. 389. M. V. Shamolin, “On a case of complete integrability in four-dimensional rigid body dynamics,” Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Vladimir, July 10–15, 2006 [in Russian], Vladimir State Univ., Vladimir (2006), pp. 226–228. 390. M. V. Shamolin, “On trajectories of characteristic points of a rigid body moving in a medium,” in: Int. Conf. “Fifth Okunev Readings,” St. Petersburg, June 26–30, 2006. Abstracts of Reports [in Russian], Balt. State Tech. Univ., St. Petersburg (2006), p. 34. 391. M. V. Shamolin, “Spatial problem on rigid body motion in a resistingmedium,” in: VIII Crimeanian Int. Math. School “Lyapunov Function Method and Its Applications,” Abstracts of Reports, Alushta, September 10–17, 2006, Tavriya National Univ. [in Russian], DiAiPi, Simpheropol’ (2006), p. 184. 392. M. V. Shamolin, “To problem on rigid body spatial drag in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 3, 45–57 (2006). 393. M. V. Shamolin, “To spatial problem of rigid body interaction with a resisting medium,” in: Abstracts of Reports of IX All-Russian Congress in Theoretical and Applied Mechanics, Nizhnii Novgorod, August 22–28, 2006, Vol. I [in Russian], N. I. Lobachevskii Nizhegodskii State Univ., Niznii Novgorod (2006), p. 120. 394. M. V. Shamolin, “Variable dissipation systems in dynamics of a rigid body interacting with a medium,” Fourth Polyakhov Readings, Abstracts of Reports of Int. Sci. Conf. in Mechanics, St. Petersburg, February 7–10, 2006 [in Russian], VVM, St. Petersburg (2006), p. 86. 395. M. V. Shamolin, “4D rigid body and some cases of integrability,” in: Abstracts of ICIAM07, Zurich, Switzerland, June 16–20, 2007, ETH, Zurich (2007), p. 311. 396. M. V. Shamolin, “A case of complete integrability in dynamics on a tangent bundle of two-dimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007). 397. M. V. Shamolin, “Case of complete integrability in dynamics of a four-dimensional rigid body in nonconcervative force field,” in: “Nonlinear Dynamical Analysis-2007,” Abstracts of Reports of Int. Congress, St. Petersburg, June 4–8, 2007 [in Russian], St. Petersburg State Univ., St. Petersburg (2007), p. 178. 398. M. V. Shamolin, “Cases of complete integrability in dynamics of a rigid body interacting with a medium,” Abstracts of Reports of All-Russian Conf. “Modern Problems of Continuous Medium Mechanics” Devoted to Memory of L. I. Sedov in Connection with His 100th Anniversary, Moscow, November 12–14, 2007 [in Russian], MIAN, Moscow (2007), pp. 166–167. 399. M. V. Shamolin, “Cases of complete integrability in dynamics of a four-dimensional rigid body in a nonconservative force field,” in: Abstract of Reports of Int. Conf. “Analysis and Singularities,” Devoted to 70th Anniversary of V. I. Arnol’d, August 20–24, 2007, Moscow [in Russian], MIAN, Moscow (2007), pp. 110–112. 400. M. V. Shamolin, “Cases of complete integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Abstracts of Reports of Int. Conf. “Classical Problems of Rigid Body Dynamics,” June 9–13, 2007 [in Russian], Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk (2007), pp. 81–82. 904
401. M. V. Shamolin, “Complete integrability of equations of motion for a spatial pendulum in medium flow taking account of rotational derivatives of moments of its action force,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 3, 187–192 (2007). 402. M. V. Shamolin, “Integrability in elementary functions of variable dissipation systems,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 38. 403. M. V. Shamolin, “Integrability of problem of four-dimensional rigid body motion in a resisting medium,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 21. 404. M. V. Shamolin, “Integrability of strongly nonconservative systems in transcendental elementary functions,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 40. 405. M. V. Shamolin, Methods of Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007). 406. M. V. Shamolin, “New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 27. 407. M. V. Shamolin, “On account of rotational derivatives of a force moment of action of the medium in angular velocity of the rigid body on body motion,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 44. 408. M. V. Shamolin, “On account of rotational derivatives of aerodynamical force moment on body motion in a resisting medium,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 26. 409. M. V. Shamolin, “On integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Modelling and Study of Systems, Sci. Conf., May 22–25, 2007. Abstracts of Reports [in Russian], Kiev (2007), p. 249. 410. M. V. Shamolin, “On integrability in transcendental functions,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 34. 411. M. V. Shamolin, “On integrability of motion of four-dimensional body-pendulum situated in over-running medium flow,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 37. 412. M. V. Shamolin, “On rigid body motion in a resisting medium taking account of rotational derivatives of aerodynamic force moment in angular velocity,” the abstract of a talk at the Workshop 905
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“Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 44. M. V. Shamolin, “On stability of a certain regime of rigid body motion in a resisting medium,” in: Abstracts of Reports of Sci. Conf. “Lomonosov Readings-2007,” Sec. Mechanics, Moscow, Moscow State Univ., April, 2007 [in Russian], MGU, Moscow (2007), p. 153. M. V. Shamolin, “On the problem of a symmetric body motion in a resisting medium,” in: Book of Abstracts of EMAC-2007 (1–4 July, 2007, Hobart, Australia), Univ. Tasmania, Hobart, Australia (2007), p. 25. M. V. Shamolin, “On work of All-Russian Conference ‘Differential Equations and Their Applications,’ Samara, June 27–July 2, 2005,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 45. M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], Ekzamen, Moscow (2007). M. V. Shamolin, “The cases of complete integrability in dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of Int. Conf. on the Occasion of the 150th Birthday of A. M. Lyapunov (June 24–30, 2007, Kharkiv, Ukraine), Verkin Inst. Low Temper. Physics Engineer. NASU, Kharkiv (2007), pp. 147–148. M. V. Shamolin, “The cases of integrability in 2D-, 3D-, and 4D-rigid body,” in: Abstracts of Short Communications and Posters of Int. Conf. “Dynamical Methods and Mathematical Modelling,” Valladolid, Spain (September 18–22, 2007 ), ETSII, Valladolid (2007), p. 31. M. V. Shamolin, “The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” in: Abstracts of 9th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2007 ), Lodz, Poland, December 17–20, 2007, Tech. Univ. Lodz (2007), p. 115. M. V. Shamolin, “The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” in: Proc. of 9th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2007 ), Lodz, Poland, December 17–20, 2007, Vol. 1, Tech. Univ. Lodz (2007), pp. 415–422. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a medium,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 17. M. V. Shamolin, “Three-parameter family of phase portraits in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 418, No. 1, 46–51 (2008). M. V. Shamolin and D. V. Shebarshov, “Projections of Lagrangian tori of a biharmonic oscillator on position state and dynamics of a rigid body interacting with a medium,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 19–23, 1997. Abstracts of Reports [in Russian], Kiev (1997), p. 142. M. V. Shamolin and D. V. Shebarshov, “Lagrange tori and the Hamilton–Jacobi equation,” in: Book of Abstracts of Conf. “Partial Differential Equations: Theory and Numerical Solutions” (Praha, August 10–16, 1998 ), Charles Univ., Praha (1998), p. 88. M. V. Shamolin and D. V. Shebarshov, “Certain problems of differential diagnosis,” in: Dynamical Systems Modelling and Stability Investigation. Sci. Conf., May 25–29, 1999. Abstracts of Reports (System Modelling) [in Russian], Kiev (1999), p. 61.
426. M. V. Shamolin and D. V. Shebarshov, “Methods for solving main problem of differential diagnosis,” Deposit. at VINITI, No. 1500-V99 (1999). 427. M. V. Shamolin and D. V. Shebarshov, “Some problems of geometry in classical mechanics,” Deposit. at VINITI, No. 1499-V99 (1999). 428. M. V. Shamolin and D. V. Shebarshov, “Certain problems of differential diagnosis,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” Fund. Prikl. Mat., 7, No. 1, 305 (2001). 429. M. V. Shamolin and D. V. Shebarshov, “Certain problems of differential diagnosis,” the abstract of a talk at the Workshop “Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 19. 430. M. V. Shamolin and S. V. Tsyptsyn, “Analytical and numerical study of trajectories of body motion in a resisting medium,” in: Scientific Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 4289, Institute of Mechanics, Moscow State Univ., Moscow (1993). 431. O. P. Shorygin and N. A. Shulman, “Entry of a disk into water at an angle of attack,” Uchen. Zap. TsAGI, 8, No. 1, 12–21 (1977). 432. S. Smale, “Rough systems are not dense,” in: A Collection of Translations, Mathematics [in Russian], 11, No. 4, 107–112 (1967). 433. S. Smale, “Differentiable dynamical systems,” Usp. Mat. Nauk, 25, No. 1, 113–185 (1970). 434. V. M. Starzhinskii, Applied Methods of Nonlinear Oscillations [in Russian], Nauka, Moscow (1977). 435. V. A. Steklov, On Rigid Body Motion in a Fluid [in Russian], Khar’kov (1893). 436. V. V. Stepanov, A Course of Differential Equations [in Russian], Fizmatgiz, Moscow (1959). 437. V. V. Strekalov, “Reflection in entrance of a disk into a water whose plane is close to the vertical plane,” Uchen. Zap. TsAGI, 8, No. 5, 66–73 (1977). 438. G. K. Suslov, Theoretical Mechanics [in Russian], Gostekhizdat, Moscow (1946). 439. E. I. Suvorova and M. V. Shamolin, “Poincar´e topographical systems and comparison systems of higher orders,” in: Math. Conf. “Modern Methods of Function Theory and Related Problems,” Voronezh, January 26–February 2, 2003 [in Russian], Voronezh State Univ., Voronezh (2003), pp. 251–252. 440. V. V. Sychev, A. I. Ruban, and G. L. Korolev, Asymptotic Theory of Separation Flows [in Russian], Nauka, Moscow (1987). 441. V. G. Tabachnikov, “Stationary characteristics of wings in small velocities under whole range of angles of attack,” in: Proc. of Central Aero-Hydrodynamical Inst. [in Russian], Issue 1621, Moscow (1974), pp. 18–24. 442. Ya. V. Tatarinov, Lectures on Classical Dynamics [in Russian], Izd. Mosk. Univ., Moscow (1984). 443. V. V. Trofimov, “Embeddings of finite groups in compact Lie groups by regular elements,” Dokl. Akad. Nauk SSSR, 226, No. 4, 785–786 (1976). 444. V. V. Trofimov, “Euler equations on finite-dimensional solvable Lie groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 5, 1191–1199 (1980). 445. V. V. Trofimov, “Symplectic structures on automorphism groups of symmetric spaces,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 6, 31–33 (1984). 446. V. V. Trofimov, “Geometric invariants of completely integrable systems,” in: Abstract of Reports of All-Union Conf. in Geometry “in the Large,” Novosibirsk (1987), p. 121. 447. V. V. Trofimov and A. T. Fomenko, “A methodology for constructing Hamiltonian flows on symmetric spaces and integrability of certain hydrodynamic systems,” Dokl. Akad. Nauk SSSR, 254, No. 6, 1349–1353 (1980). 907
448. V. V. Trofimov and M. V. Shamolin, “Dissipative systems with nontrivial generalized Arnol’d–Maslov classes,” the abstract of a talk at the Workshop in Vector and Tensor Analysis Named after P. K. Rashevskii, Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 62 (2000). 449. Ch. J. de la Vall´ee Poussin, Lectures on Theoretical Mechanics, Vols. I, II [Russian translation], Inostr. Lit., Moscow (1948, 1949). 450. L. E. Veselova, “On dynamics of a body with ellipsoidal hole filled with a fluid,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 64–67 (1985). 451. S. V. Vishik and S. F. Dolzhanskii, “Analogs of Euler–Poisson equations and magnetic electrodynamics related to Lie groups,” Dokl. Akad. Nauk SSSR, 238, No. 5, 1032–1035. 452. I. N. Vrublevskaya, “On geometric equivalence of trajectories and semitrajectories of dynamical systems,” Mat. Sb., 42 (1947). 453. I. N. Vrublevskaya, “Some criteria of equivalence of trajectories and semitrajectories of dynamical systems,” Dokl. Akad. Nauk SSSR, 97, No. 2 (1954). 454. C. L. Weyher, “Observations sur le vol plane par obres,” L’Aeronaute (1890). 455. E. T. Whittecker, Analytical Dynamics [Russian translation], ONTI, Moscow (1937). 456. M. V. Yakobson, “On self-mappings of a circle,” Mat. Sb., 85, 183–188 (1975). 457. N. E. Zhukovskii, “On a fall of light oblong bodies rotating around their longitudinal axis,” in: A Complete Collection of Works [in Russian], Vol. 5, Fizmatgiz, Moscow (1937), pp. 72–80, 100–115. 458. N. E. Zhukovskii, “On bird soaring,” in: A Complete Collection of Works [in Russian], Vol. 5, Fizmatgiz, Moscow (1937), pp. 49–59. 459. Yu. F. Zhuravlev, “Submergence of a disk in a fluid at an angle to a free surface,” in: A Collection of Works in Hydrodynamics [in Russian], Central Aero Hydrodynamical Institute, Moscow (1959), pp. 164–167. 460. V. F. Zhuravlev and D. M. Klimov, Applied Methods in Oscillation Theory [in Russian], Nauka, Moscow (1988). 461. N. A. Zlatin, A. P. Krasil’shchikov, G. I. Mishin, and N. N. Popov, Ballistic Devices and Their Applications in Experimental Studies [in Russian], Nauka, Moscow (1974). M. V. Shamolin Lomonosov Moscow State University, Moscow, Russia E-mail:
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