Insights into the algebraic structure of Lorenz-like ... - Semantic Scholar

CHAOS 17, 023104 共2007兲

Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models Christophe Letellier CORIA UMR 6614—Université de Rouen, Avenue de l’Université, Boite Postale 12, F-76801 Saint-Etienne du Rouvray Cedex, France

Gleison F. V. Amaral Universidade Federal de São João del-Rei, Pça Frei Orlando 170, 36307-352 São João del-Rei, M.G., Brazil

Luis A. Aguirre Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, M.G., Brazil

共Received 19 October 2006; accepted 22 January 2007; published online 18 April 2007兲 The characterization of chaotic attractors has been a widely addressed problem and there are now many different techniques to define their nature in a rather accurate way, at least in the case of a three-dimensional system. Nevertheless, the link between the structure of the ordinary differential equations and the topology of their solutions is still missing and only a few necessary conditions on the algebraic structure are known today. By using a feedback circuit analysis, it is shown that it is possible to identify the relevant terms of the equations, that is, the terms that really contribute to the structure of the phase portrait. Such analysis also provides some guidelines for constructing piecewise affine models. Moreover, equivalence classes can be defined on the basis of the active feedback circuits involved. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2645725兴 It has been known since Poincaré’s early works that phase portraits—chaotic attractors in the case considered here—are structured around the fixed points of the system. It is known here that these fixed points are not necessarily sufficient to fully understand such structures and that other types of singularities, namely double nullclines, must be considered too. Several Lorenz-like attractors are considered and classified into two groups according to their singularities and the algebraic structures of the underlying ordinary differential equations. The latter are investigated with the help of a feedback circuit description allowing a decomposition of the whole system into subsystems.

I. INTRODUCTION

The properties of the phase portraits induced by nonlinear dynamical systems have been widely investigated. This was in fact the program left by Poincaré when he realized that the three-body problem does not have any general analytical solution.1 The main reason chaotic behaviors are investigated mainly in phase space is that the presence of a single nonlinearity is often sufficient to prevent any analytical computation. There are some ways to circumvent such a problem. For instance, numerical computations 共simulations兲 are very useful in understanding how the system evolves in phase space, although they do not usually provide any analytical solution. Another possible approach, which does permit some analytical exercise, is to transform the nonlinear system into a linear one. In the latter case, however, the results can only be of a local character, and a more global approach is still missing. 1054-1500/2007/17共2兲/023104/11/$23.00

The idea to approximate a nonlinear dynamical system by a piecewise affine model is not new and was used, for instance, by Lozi2 to approximate the Hénon map.3 Approximations to continuous nonlinear dynamical systems were also considered, but the number of subsystems involved was too large to be of any interest for any analytical study.4–6 This paper is concerned mainly with the global analysis of continuous autonomous nonlinear dynamical systems. In order to achieve this, we employ the concept of feedback circuits, which are a tool that in many cases will provide a rough idea of how the dynamics and the system equations are linked. Also, a procedure to build piecewise affine 共or switched兲 models will be used. The aim with this is twofold. First, being a method for synthesis, it is used as a means to check the main conclusions arrived at through feedback circuit analysis. Second, switched models have attracted a great deal of attention recently, not only as a means to build devices with complex dynamics from simple elements,7 but also as a way to directly applying results that are available either for linear or switched systems.8 See Ref. 9 and references therein for other examples. Finally, the Lorenz system was chosen as a benchmark. A first study that considered mainly the Rössler system and gave some hints as to how to address the Lorenz system was presented in Ref. 10. It seems important to mention that to start from a switched model, such as the Matsumoto-Chua circuit, would hinder us from assessing the full procedure because the concept of feedback circuit analysis was developed to deal with nonswitched systems. In the remainder of this Introduction, some background and some basic references on the main parts of the paper will be provided.

17, 023104-1

© 2007 American Institute of Physics

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-2

Chaos 17, 023104 共2007兲

Letellier, Amaral, and Aguirre

Since the seminal paper by Lorenz back in 1963,11 Lorenz-like dynamics were identified in different physical systems as in hydrodynamics, magnetohydrodynamics, lasers, etc. An extended investigation of the underlying dynamics of the Lorenz system was given by Sparrow in terms of periodic orbits and symbolic dynamics.12 A first topological analysis was provided by Birman and Williams, using the concept of a knot-holder, commonly called a template.13 The Lorenz system has a rotation symmetry around the z axis. This symmetry can be conveniently taken into account to improve the understanding of its dynamics.14 The role played by this symmetry is not so trivial, and it may affect the observability of the system depending on which variable共s兲 is 共are兲 measured.15,16 When the Lorenz dynamics is investigated by means of a Poincaré section, it immediately becomes obvious how the symmetry complicates the analysis. In the case of a system with symmetry properties, it was shown that the Poincaré section is in fact the union of many Poincaré surfaces, one being related to the others by the symmetry. For instance, the number of different surfaces is related to the order of the symmetry.17 This received a more general formulation in the context of the bounding tori, which are closed surfaces surrounding the attractor.18 Basically, a bounding torus has a genus g equal to the number of fixed points surrounded by the flow. Saddles are distinguished from foci. The Poincaré section is the union of 共g − 1兲 Poincaré surfaces. Recently, a piecewise affine model was proposed for the Lorenz system by the means of two linear subsystems based on the Jacobian matrix estimated at each of the saddle foci.10 Basically, an affine system was associated with each fixed point of the focus type. Such affine systems were built using the Jacobian matrix of the system evaluated at the corresponding fixed point. As mentioned in the context of the bounding tori, the saddle fixed points have to be distinguished from the focus fixed points. It was thus shown that a switching surface was sufficient to take into account the role played by the saddle fixed point located at the origin of the phase space of the Lorenz system.10 The chaotic attractors solution to the piecewise affine model were thus topologically equivalent to those solutions to the original Lorenz system. One of the great advantages of the piecewise affine models built according to our procedure is that they convert a nonlinear dynamical system into a switched system with a small number of linear subsystems. This feature turns out to be very useful because it makes it possible to return to analytical computations that will be useful for investigating the link between the algebraic structure of the ordinary differential equations and the topological properties of the chaotic attractor in the phase space. Some effort has been devoted to the qualitative analysis of dynamical systems. This is the so-called feedback circuit analysis as introduced by Thomas 共see Refs. 19 and 20 and references therein兲. It is based on an analysis of the signs of the products occurring in the determinant of the Jacobian matrix of the system. It constitutes an original and powerful approach to investigate the algebraic structure of the ordinary differential equations in relation to the topology of their corresponding phase portraits. In such an approach, the dy-

namics of a nonlinear dynamical system is considered as resulting from the interactions of full feedback circuits, one being associated with each product of the Jacobian of the system studied. In this paper, nine Lorenz-like systems will be investigated using the feedback circuit analysis in order to determine the relevant terms on the right-hand side of the corresponding ordinary differential equations. In particular, it will be shown how they contribute to the topology of the phase portrait. Moreover, these nine systems share the same description in terms of an active feedback circuit, which can therefore be used to define an equivalence class. Meanwhile, such an analysis provides guidelines for constructing piecewise affine models. The rest of the paper is organized as follows. Section II is devoted to a brief introduction of the feedback circuit description with the Lorenz treated as an example. Section III describes the algebraic structure of the nine Lorenz-like systems in terms of feedback circuits. Section IV addresses the problem of constructing piecewise affine models from a feedback circuit analysis, and Sec. V presents a conclusion. II. FEEDBACK CIRCUIT ANALYSIS

The ideas of feedback circuits were developed in the context of dynamical systems with linear step-function dynamics.21,22 Such an analysis was then extended to more general dynamical systems.19,20 It provides a good starting point to investigate how the algebraic structure of ordinary differential equations is related to their asymptotic behaviors. In this spirit, the analysis of some simple chaotic flows brought to light the important role the double nullcline can have on the topology of the attractors.19,23 Feedback circuits can thus serve as a qualitative guide to interpret differential equations, at least in some cases. Let us start with a dynamical system x˙i = f i共x1,x2,x3兲

with i = 1,2,3

共1兲

described in a three-dimensional phase space for the sake of simplicity. The interactions between the dynamical variables xi can be defined using the elements of the Jacobian matrix. Variable x j acts on variable xi when the element Jij of the Jacobian matrix is nonzero. This action is positive or negative depending on the sign of element Jij. A full circuit is defined as a sequence of nonzero elements in the Jacobian matrix corresponding to one of the products appearing in the analytic expression of its determinant. For a threedimensional system, the Jacobian reads det共J兲 = J11J22J33 − J11J23J32 − J22J13J31 − J33J12J21 + J12J23J31 + J13J32J21

共2兲

and up to six full circuits can be identified. Partial feedback circuits that do not involve all dynamical variables are also of interest. Thus, for three-dimensional systems, a partial circuit is defined by a two-element product JijJ ji 共i ⫽ j兲 or by a single element Jkk. The six full circuits identified for three-dimensional systems can be classified into two groups. The first group is decomposable while the second is not.

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-3

Chaos 17, 023104 共2007兲

Structure of Lorenz-like systems

• Decomposable. The full circuit is the union of partial circuits. It may be either the union of one two-element circuit JijJ ji 共i ⫽ j兲 and one single-element circuit Jkk 共k ⫽ i and k ⫽ j兲, or the union of three one-element circuits, Jkk. • Nondecomposable. The full circuit cannot be decomposed into the union of partial circuits. For a three-dimensional system, we therefore have four decomposable circuits 共J11J22J33, J11J23J32, J22J31J13, and J33J12J21兲 and two nondecomposable circuits 共J12J23J31 and J13J32J21兲. Each full circuit, when isolated, can produce one or more fixed points. For instance, the two indecomposable full circuits, J12J23J31 and J13J32J21, are associated with eigenvalues ␭k = 冑3 J12J23J31 and ␭k = 冑3 J13J32J21, respectively. They can therefore be associated with saddle foci, of type SF+ 共one positive real eigenvalue and a pair of complex-conjugate eigenvalues with negative real parts兲 when JijJ jkJki ⬎ 0, or SF− 共one negative real eigenvalue and two complex-conjugate eigenvalues with positive real parts兲 when JijJ jkJki ⬍ 0. The three decomposable full circuits JijJ jiJkk have eigenvalues computed using elements Jij, J ji, and Jkk completed by the diagonal elements Jii andJ jj. These eigenvalues are

␭1,2 =

共3兲

In the case above, the associated fixed point can be a node, a node focus, a saddle focus 共SF+ or SF−兲, or a saddle. In the case of circuit JiiJ jjJkk, the eigenvalues are the diagonal elements themselves. All these elements are real and the associated fixed point is a node or a saddle point. When the system is linear, all the elements of the Jacobian matrix are constant and, consequently, a single fixed point can be associated with each full circuit. In the nonlinear cases, the values of the Jacobian matrix vary through phase space, which can be partitioned into different domains in which the circuits have the same types of eigenvalues.

The case of the Lorenz system

The Lorenz system is now used as an example of such a feedback circuit analysis. The Lorenz system11 x˙ = − ␴x + ␴y,

y˙ = Rx − y − xz,

Circuit J11J22J33 = −␴b ␭1 = −␴ ␭2 = −1 ␭3 = −b

Stable node Circuit J11J23J32 = + ␴x2

␭1 = −␴ ␭2,3 =

−共b+1兲±冑b2+2b−4bR+1

Stable node

2

Circuit J33J12J21 = −b␴共R − z兲

冑

−共␴+1兲± 共␴+1兲2+4␴共R−1−z兲

␭1,2 = ␭3 = −b

Saddle for z = 0

2

Circuit J12J23J31 = −␴xy

␭1,2,3 = 冑3 −␴xy

saddle focus

冦 冨

共4兲

冦

x0 = 0

F0 = y 0 = 0 z0 = 0

共Jii + J jj兲 ± 冑共Jii − J jj兲2 + 4JijJ ji , 2

␭3 = Jkk .

TABLE I. The four full feedback circuits identified from the Jacobian of the Lorenz system. The analytic expressions of the eigenvalues associated with each circuit 共when isolated兲 are reported. The type of fixed point associated with each circuit corresponds to the usual parameter values leading to the well-known Lorenz attractor.

x± = ± 冑b共R − 1兲

F± = y ± = ± 冑b共R − 1兲 兩

and

共5兲

z± = R − 1.

For 共R , ␴ , b兲 = 共28, 10, 8 / 3兲, the asymptotic behavior is chaotic. F0 is a saddle and F± are two saddle foci related by a rotation symmetry around the z axis. The Jacobian matrix of the Lorenz system reads

冤

−␴

+␴

0

冥

J= R−z −1 −x . y x −b

共6兲

Among the six possible circuits contributing to its determinant, only four full feedback circuits are identified since the other two are zero due to J13 = 0. The four circuits and their corresponding eigenvalues are reported in Table I. Among them, circuit J11J22J33 and circuit J11J23J32 correspond to a stable fixed points. There is no such fixed point for the parameter values associated with a chaotic attractor. These circuits are therefore not active, that is, they are not relevant for the topology of the phase portrait. Only the two remaining circuits could potentially contribute in a significant way to the topology of the Lorenz system. The algebraic sign of the circuit J12J23J31 = −␴xy depends on the location in the phase space. Its associated eigenvalues are computed from the Jacobian matrix only made of J12, J23, and J31, that is,

J=

冤

·

+␴

·

·

+y

·

·

冥

−x . ·

共7兲

They are z˙ = − bz + xy has three fixed points with coordinates as follows:

␭1,2,3 = 冑− ␴xy. 3

共8兲

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-4

Chaos 17, 023104 共2007兲

Letellier, Amaral, and Aguirre

TABLE II. Cœoefficients of the nine quadratic systems with a rotation symmetry around the z axis. The first five systems have three fixed points, one located at the origin of the phase space and two that are related by the rotation symmetry. The last four systems have only the two symmetry-related fixed points.

共1兲 共2兲 共3兲 共4兲 共5兲 共6兲 共7兲 共8兲 共9兲

System

x a1

y a2

yz a3

x b1

y b2

xz b3

— c0

z c1

x2 c2

xy c3

y2 c4

Reference

Lorenz Chen and Ueta Wang, Singer, and Bau Shimizu and Morioka Rucklidge Burke and Shaw Sprott B Sprott C Rikitake

−␴ −␴ −␴ 0 0 −S −␴ −␴ −␮

+␴ +␴ ␴ +1 +1 +S +␴ +␴ 0

0 0 0 0 0 0 0 0 +1

R R−␴ 0 +1 −␭ 0 0 0 −␣

−1 R −1 −␮ +␬ −1 0 0 −␮

−1 −1 −1 −1 −1 +R +1 +1 +1

0 0 Ra 0 0 V b b +1

−b −b −1 −␣ −1 0 0 0 0

0 0 0 +1 +1 0 0 0 0

+1 +1 +1 0 0 −S −1 0 −1

0 0 0 0 0 0 0 −1 0

11 25 26 27 28 29 30 30 31

They correspond to a saddle focus SF− when xy is positive, that is, in the domain of the phase space containing the two fixed points F±. The circuit J12J23J31 is thus active in these two domains. A simple partition of the phase space would be a plane defined by y = −x. All planes containing the z axis orientated with an angle less than ␲2 with the y axis would be appropriate too. The circuit J33J12J21 = −b␴共R − z兲 has a product with a sign that depends on the location in the phase space. When estimated at the fixed point located at the origin, its eigenvalues correspond to a saddle. It is therefore associated with the fixed point F0. It is thus active. The sign of its product J33J12J21 leads to a partition of the phase space by the plane z = R, a partition that does not bring any obvious understanding of the structure of the Lorenz attractor. Since the circuit J33J12J21 is decomposable, it is also possible to only consider the partial circuit J12J21 as follows. The eigenvalues of the partial circuit J12J21 are computed from the 2 ⫻ 2 Jacobian matrix

J=

冋

−␴

+␴

R−z −1

册

共9兲

which only involves variables x and y. The eigenvalues have the form

␭1,2 =

− 共␴ + 1兲 ± 冑共␴ − 1兲2 + 4␴共R − z兲 4␴

共10兲

and correspond to stable foci when z ⬎ R + 关共␴ − 1兲2 / 4␴兴共=30.025兲 and to saddles when z ⬍ R − 1共=27兲 for the common parameter values. This partial circuit involves the x and y dynamical variables. It corresponds to the double nullcline nullx 艚 nully obtained by setting to zero their derivatives, that is, x˙ = 0 = − ␴x + ␴y, 共11兲 y˙ = 0 = Rx − y − xz.

The double nullcline is thus defined by x = y = 0 for any z value. It thus corresponds to the z axis. Note that the saddle F0 belongs to the z axis. This double nullcline nullx 艚 nully is therefore more relevant than the fixed point F0, as shown below. Since, for the common values, the attractor is quite far from the z axis when z ⬎ R − 1 共see the next section兲, the attractor is mainly influenced by the segment of the z axis with a transverse manifold made of saddles. It was shown in a previous investigation24 that the topology of the Lorenz attractor was mainly induced by the two saddle foci F± and the z axis. From the present feedback circuit analysis, it was thus possible to find these relevant ingredients. The Lorenz system is therefore characterized by the full circuit J12J23J31 responsible for the two symmetry-related saddle foci F± and the partial circuit J12J21 associated with the double nullcline nullx 艚 nully defined as the z axis.

III. RELEVANT SINGULAR SETS FOR LORENZ-LIKE ATTRACTORS

It was recently shown that ten chaotic systems belong to the topological class for which the Lorenz system can be considered as the emblem.24 The equations of these systems are reported in Table II. All these systems are quadratic with at least five terms on the right-hand side of the equations. Some of them have been rewritten with a permutation between the variables in such a manner that all of them are invariant under a rotation symmetry Rz共␲兲 around the z axis. They have at least two fixed points F± related by the rotation symmetry. The Burke and Shaw system has been rewritten under a reflection symmetry 共x 哫 −x兲 to have the same orientation in the phase space as most of the others, that is, the two fixed points F± are located on the first bisecting line of the x-y plane. Only the Shimizu-Morioka and the Rucklidge systems have their fixed points located along the x axis 共Fig. 1兲. In all cases, the two fixed points F± are saddle foci SF− as for the Lorenz system. All these systems have two active feedback circuits. There is one full circuit whose eigenvalues correspond to saddle foci SF− when they are estimated at the fixed points F±. For all the systems, excepted the Sprott C system 共later

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-5

Chaos 17, 023104 共2007兲

Structure of Lorenz-like systems

␭1,2,3 = 冑J12J23J31 3

FIG. 1. Projections in the x-y plane of the chaotic attractors solution to the five systems with three fixed points.

discussed in detail兲, the active full circuit is the nondecomposable circuit J12J23J31. When estimated at the fixed points F±, the product J12J23J31 is always negative as required to have eigenvalues

共12兲

corresponding to a saddle focus SF−. For all the systems, the second active circuit is the partial circuit J12J21, which always depends on the z variable 共Table III兲. As for the Lorenz system, the partial circuit J12J21 defines a double nullcline nullx 艚 nully corresponding to the z axis. As for the Lorenz system, the z axis always has a transverse manifold made of foci, node, and saddles depending on the z value 共according to the dependence on the z value of the product J12J21兲. Consequently, as for the Lorenz system, the two relevant singular sets for the structure of the phase portraits induced by these systems are the two symmetry related saddle foci F± and the double nullcline nullx 艚 nully, which corresponds to the z axis. For the first five systems, there is a third fixed point F0 located at the origin of the phase space. It always belongs to the double nullcline nullx 艚 nully and is not the most relevant singular set for the topology of the phase portrait. As shown in Figs. 1 and 2, these nine systems produce two types of chaotic attractors. The first type corresponds to the attractor sometimes called the Burke and Shaw attractor.32 Such attractors are only made of foldings24 and are surrounded by a genus-1 bounding torus. Their firstreturn maps to a Poincaré section are everywhere differentiable. The parameter values reported in Figs. 1 and 2 for the genus-1 attractor correspond to a nearly complete symbolic dynamics 共all possible symbolic sequences are realized within the attractor兲 as described in Ref. 32 for the Burke and Shaw system. This symbolic dynamics is constructed with four symbols as suggested by the first-return maps, which are made of four monotonic branches.32 The single hole of the genus-1 bounding torus corresponds to a focus.18 The second type of attractor shown in Figs. 1 and 2 corresponds to the Lorenz attractor that is surrounded by a genus-3 bounding torus. Two holes surround the two symmetry-related fixed points F±. From the bounding tori description, the hole in the middle of the attractor 共in fact the z axis兲 must correspond to a saddle.18 The parameter values reported in Figs. 1 and 2 are chosen to have a first-return map with a nondifferentiable critical point 共a cuspide兲 corresponding to a two-symbol dynamics nearly complete. According to the bounding tori description, the main departure between these two types of attractors is that the genus-1 attractor has to be structured around a focus while the genus-3 attractor has to be organized around a saddle. From the feedback circuit analysis, only the double nullcline nullx 艚 nully—the z axis—can explain such a change. According to the previous analysis on the Lorenz system, the z axis has a transverse manifold of saddles, node, or foci as reported in Table IV. The nature of the attractor thus depends on its location around the z axis and on the closeness of the trajectory to the different segments of the z axis. For both types of attractors, the minimum distance dmin 共relative to the size of the attractor兲 between the attractors and the z axis was computed with their corresponding z value, zd. Moreover, the relative distance between the attractor and the z axis was also computed on the plane z = zc = R + 关共␴1兲2 / 4␴兴. For the genus-1 attractor,

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-6

Chaos 17, 023104 共2007兲

Letellier, Amaral, and Aguirre

TABLE III. Products of the Jacobian associated with each feedback circuit for the nine Lorenz-like systems investigated here. The first five systems have three fixed points, two saddle foci F± and one saddle F0. The four others have only the two saddle foci F±. Product Lorenz Chen and Ueta Wang, Singer, and Bau Shimizu and Morioka Rucklidge Burke and Shaw Sprott B Sprott C Rikitake

J11J22J33

J11J23J32

J22J13J31

J33J12J21

J12J23J31

J13J32J21

−␴b ␴bR −␴ 0 0 0 0 0 0

␴x2 ␴x2 ␴x2 0 0 S 3x 2 ␴x2 +2␴xy ␮x2

0 0 0 0 0 0 0 0 ␮y2

−b␴共R − z兲 −b␴共R − ␴ − z兲 ␴z −␣共1 − z兲 ␭+z 0 0 0 0

−␴xy −␴xy −␴xy −2x2 −2x2 S3xy −␴xy 0 −xyz

0 0 0 0 0 0 0 0 xy共a + z兲

the minimum distance dmin is observed close to the plane z = R − 1 and remains more than 1% of the attractor size. Meanwhile, the attractor is not so far from the foci 共4.5% of the attractor size in the plane z = zc兲. This means that the upper part of the attractor is quite influenced by the transverse stability of the z axis corresponding to foci 关Fig. 3共a兲兴. Contrary to this, for the genus-3 attractor, the minimum distance between the attractor and the z axis is very small 共0.18% of the attractor size兲 and in a plane 共z = 12.8兲 clearly below the threshold where the transverse stability switches from a saddle type to a node type 关Fig. 3共b兲兴. As seen in the latter figure, the attractor is close to the z axis when its transverse stability corresponds to a saddle and quite far 共more than 10% of the attractor size兲 from the z axis with a focus transverse stability. This is therefore consistent with the bounding tori description. For each type of attractor, the vector field of the Lorenz system is plotted on the plane z = zd corresponding to the minimum distance dmin and on the plane z = zc 共Fig. 4兲. For both cases, the vector field evidences a transverse stability to the z axis corresponding to a saddle for z = zd and to a focus for z = zc. The stable eigendirections of the saddle defines the separatrix between a reinjection of the trajectory in the same wing and a transition from one wing to the other. The main difference between the two cases shown in Fig. 4 is the orientation of these stable eigendirections. Such a feature will be of importance when piecewise affine models are discussed in the next section. All diagonal elements of the Jacobian matrices of all the systems are constant. The fixed point F0 has been shown to be a nonrelevant singular set for the topology of the Lorenzlike attractors. This is why topologically equivalent attractors can be found, even in systems not having this fixed point. On the other hand, the systems must have the two saddle foci related by the rotation symmetry around the z axis and a z axis with a transverse stability that switches from a saddle to a node and then to a focus. Such singular sets are therefore dominant. It can be observed that all the systems presenting the fixed point F0 have the decomposable full circuit J33J12J21, which always depends on the z variable. Since the partial circuit J12J21 is identified in all systems, only the lack of element J33 is responsible for the absence of the fixed point

F0 共this can be checked in Table II兲. The nine cases considered here have at least two nonlinearities. All of them have the term xz in the second equation and one of the three quadratic possibilities permitted by the rotation symmetry Rz共␲兲, that is, x2, xy, or y 2, can be used in the third equation. The term y 2 is only used in the Sprott C system, which is minimal in the sense that it has only five terms on the righthand side of the equations. It was shown that the simplest equivariant system with five terms but a single nonlinearity has an inversion symmetry and not a rotation symmetry.33 The Sprott C system could therefore be the minimal—from the algebraic point of view—equivariant system with a rotation symmetry. From the feedback point of view, the Sprott C system is also minimal since it has only a single full circuit 共J11J23J32兲. For these reasons, it is not surprising that some difficulties were encountered in observing an attractor solution to the Sprott C system with a first-return map only made of two monotonic branches split by a cuspide as a critical point like the original Lorenz map. Indeed, it is known that minimal systems often have a small domain of the parameter space over which a chaotic attractor is observed.34 A similar comment can be made for the Sprott B system, which has also five terms but two full feedback circuits 共J12J23J31 and J11J23J32兲, a single one being active. Among the five systems with three fixed points, the Shimizu-Morioka and the Rucklidge systems are those that have only two full circuits while the three others have four 共Table III兲. These are the only two to have the three fixed points located along the x axis. First, only these two circuits confirm the fact that additional nonactive full circuits are not relevant for the topology of the attractor. We could conjecture that the nonactive full circuits 共J11J22J33 and J11J23J32兲 only induce rotation in the phase space 共to locate the fixed points along the first bisecting line兲 or eventually a rescaling as observed in the Chen-Ueta system 关Fig. 1共b兲兴. The fact that some terms were not important for the topology of the attractor was already mentioned by Lainscsek35 in the context of global modeling using an ansatz library.36 The feedback circuit analysis is thus helpful to understand the role the algebraic structure of the equations may have on the topology of the phase portrait.

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-7

Chaos 17, 023104 共2007兲

Structure of Lorenz-like systems

TABLE IV. Types of the transverse manifold of the z axis depending on the z value for the Lorenz system. Genus-1 R = 278.56 ␴ = 30, b = 1

Genus-3 R = 28 ␴ = 10, b = 8 / 3 Saddles

z = R − 1 = 277.56

z = R − 1 = 27 Nodes

z=R+

共␴−1兲2 4␴

= 285.56

z=R+

共␴−1兲2 4␴

= 30.025

Foci

associate one affine subsystem with each fixed point of the focus type, the subsystems being linked by switching surfaces. In the Lorenz case, it was found that the fixed point of the saddle type did not need to be associated with an affine subsystem and that its role, which is to separate the two foci,

FIG. 2. Projections in the x-y plane of the chaotic attractors solution to the four systems with two fixed points. In the case of the Sprott B and C systems, the genus-3 attractor does not correspond exactly to an attractor with only a tearing mechanism, that is, characterized by a first-return map only made of two monotonic branches split by a nondifferentiable critical point 共a cuspide兲 as the original map computed by Lorenz in 1963.

IV. GUIDELINES FOR BUILDING PIECEWISE AFFINE MODELS

In a recent study, a new procedure to construct piecewise affine models was introduced.10 Basically, the idea was to

FIG. 3. Projection in the x-z plane of the types of chaotic attractors solution to the Lorenz system. The transverse stability of the z axis is reported versus the z value. The minimum distance dmin between the attractor and the z axis is also indicated as well as the distance at z = zc. 共a兲 Genus-1 attractor and 共b兲 genus-3 attractor. Parameter values are given in Table IV.

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-8

Letellier, Amaral, and Aguirre

Chaos 17, 023104 共2007兲

FIG. 4. Vector field of the Lorenz system in two planes transverse to the z axis. Depending on the z value, a saddle or a focus is identified on the z axis. The eigendirections are indicated by a solid line 共stable manifold兲 and a dashed line 共unstable manifold兲. Cases of genus-1 attractor projected onto the plane defined by 共a兲 z = zd and 共b兲 z = zc, and cases of genus-3 attractor projected onto the plane defined by 共c兲 z = zd and 共d兲 z = zd.

was reproduced by the switching surface. From the previous feedback circuit analysis, it has been shown that the saddle fixed point F0 was not a relevant ingredient for the topology of the Lorenz attractor. It thus provides a good justification of the way the piecewise affine model of the Lorenz attractor was previously constructed. Moreover, by plotting the vector field in a plane transverse to the rotation axis 共Fig. 4兲, it was shown that the main departure between the genus-1 and the genus-3 attractors was the orientation of the switching surface, that is, the stable manifold of the saddle associated with the transverse stability of the double nullcline nullx 艚 nully 共the z axis兲. According to the first piecewise affine model10 and the previous feedback circuit analysis, a piecewise model for the Lorenz attractor is based on two linear subsystems, each one

built from the Jacobian matrix of the Lorenz system estimated at each of the two symmetry-related saddle foci F±. The switching surface must be the separatrix between the domains of influence of each fixed point, that is, roughly, a plane defined by y = −x as suggested by the product J12J23J31 = −␴xy 共each fixed point is located in such a way that xy ⬎ 0兲. The switching line is thus mainly orthogonal to the first bisecting line where the two fixed points F± are located. In order to preserve the symmetry properties, the switching surface has to contain the rotation axis. From our previous analysis, the orientation of the switching surface can be an important parameter. In the piecewise model, the switching surface will thus depend on the angle ␪ between the switching surface and the x axis. The piecewise affine model thus reads

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-9

冦

冤 冤冥 冤 x˙ y˙ = z˙

Chaos 17, 023104 共2007兲

Structure of Lorenz-like systems

−␴

+␴

·

R − z+ − 1 − x+ y+

+ x+ − b

−␴

+␴

·

R − z− − 1 − x− y−

+ x− − b

冥冤 冥 冥冤 冥 x − x+

y − y+

if x tan ␪ + y ⬎ 0,

z − z+

x − x−

y − y−

if x tan ␪ + y ⬍ 0,

z − z−

where ␪ has to be within the interval 关0 ; ␲2 兴. Since there is a new parameter in the piecewise model 共13兲 that is the angle ␪, the solution of this model will rarely correspond exactly to those observed with the original system 共with the same parameter values兲. The parameter values have to be adjusted to balance the effect of the displacement in the parameter space necessarily induced by the introduction of the switching surface as well as the linearization. A similar effect was observed when discretization of ordinary differential equations was performed using nonstandard scheme.37 As a matter of fact, two chaotic attractors topologically equivalent to the attractor solution to the original Lorenz system were obtained 共Fig. 5兲. The genus-3 attractor was obtained very easily over a wide interval of ␪ values 关Fig. 5共b兲兴. The R-parameter value was adjusted to produce a well developed chaos. The genus-1 attractor was only observed for a rather limited range of ␪ values. Moreover, the appropriate ␪ value for the genus-1 attractor should be slightly smaller than for the genus-3 attractor as suggested by the vector field of the Lorenz system 共Fig. 4兲. It seems that the piecewise model is quite fragile around the switching surface and the time step was decreased to 2 ⫻ 10−3 to avoid numerical instabilities. The main reason that could explain such a lack of robustness is related to the fact that the genus-1 attractor is structured by a focus around the z axis, a component that is not introduced in the piecewise affine model 共13兲. Adding a third subsystem to this piecewise affine model is beyond the scope of this paper and is postponed for future works. In a previous study, a genus-1 attractor was obtained with a switching surface with ␪ = 0.68␲. Such a surface is not in agreement with the conclusions obtained from the feedback circuit analysis since it is roughly in the middle of the domains of influence of fixed points F±. It is therefore not surprising that the shape of the attractor is quite different from those expected 关compare Fig. 1共a兲 with Fig. 6兴. The last example shows that the orientation of the switching surface acts as a bifurcation parameter. The feedback circuit analysis therefore helped us to better design the piecewise affine model.38

冧

共13兲

from a full nondecomposable circuit. These two fixed points are responsible for the oscillating properties of the Lorenz system. Another singular set important for the topology of the Lorenz attractor is not the saddle located at the origin of the phase space but a double nullcline that corresponds in fact to the rotation axis. This axis was found to be made of segments associated with different transverse stability. When the transverse manifold corresponds to a saddle, a tearing mechanism is induced, leading to a chaotic attractor topo-

V. CONCLUSION

By applying a feedback circuit analysis, we were able to identify by analytical computations the relevant singular sets contributing to the topology of the Lorenz attractor. It was thus shown that the two symmetry-related saddle foci result

FIG. 5. Chaotic attractors solution to the piecewise model 共13兲 for the Lorenz system. 共a兲 Genus-1 and 共b兲 genus-3 attractors were obtained with an appropriate orientation of the switching surface. Parameter values 共a兲 R = 254, ␴ = 30, b = 1, ␪ = 0.4355␲, and 共b兲 R = 55, ␴ = 10, b = −8 / 3, ␪ = 0.5␲.

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-10

Chaos 17, 023104 共2007兲

Letellier, Amaral, and Aguirre 1

FIG. 6. Chaotic attractor solution to the piecewise model 共13兲 with a switching surface close the first bisecting line 共␪ = 0.68␲兲, a property not corresponding to the actual structure of the dynamics. Parameter values: R = 28, ␴ = 10, b = 8 / 3.

logically equivalent to the common Lorenz attractor observed with the original parameter values chosen by Lorenz. When the parameter values are different, it has been shown that the location of the attractor may be in such a way that it is closer to the segment with a focus as a transverse manifold. The double nullcline thus induces a folding, leading to a very different attractor that is topologically equivalent to the attractor originally observed in the Burke and Shaw system. The feedback circuit analysis provides an accurate description of this feature. Nine quadratic systems have been investigated. All of them share a rotation symmetry around the z axis, two symmetry-related saddle foci, a double nullcline—in fact associated with the rotation axis—made of three segments with a transverse stability being associated with a saddle, a node, and a focus, respectively. All these properties are encoded into two feedback circuits that define an equivalence class for the Lorenz-like systems. We thus showed that particular topological structure are induced by particular elements of the Jacobian matrix of the system. For instance, from the nine systems investigated, it was deduced that the presence of the saddle at the origin of the phase space requires the presence of the element J33 in the Jacobian matrix. On the other hand, the elements corresponding to the active circuits 共full or partial兲 are important for the topology of the phase portrait while the others should only contribute to its orientation and its shape. The feedback circuit analysis also provides some guidelines to build piecewise switching models. The model previously obtained was thus improved. Other applications of the feedback circuit analysis should appear soon in control theory, embedology, or observability-related problems.

ACKNOWLEDGMENTS

C.L. wishes to thank R. Thomas and M. Kaufmann for helpful and stimulating discussions. This work was partly supported by CNRS and CNPq.

H. Poincaré, “Sur le problème des trois corps et les équations de la dynamique,” Acta Math. 13, 1–279 共1890兲. 2 L. Lozi, “Un attracteur étrange du type attracteur de Hénon,” J. Phys. 共Paris兲, Colloq. 39, C5–C9 共1978兲. 3 M. Hénon, “A two-dimensional mapping with a strange attractor,” Commun. Math. Phys. 50, 69–77 共1976兲. 4 H. L. Hiew and C. P. Tsang, “An adaptive fuzzy system for modeling chaos,” Inf. Sci. 共N.Y.兲 81, 193–212 共1994兲. 5 J. D. Farmer and J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845–848 共1987兲. 6 M. Storace and O. De Feo, “Piecewise-linear approximation of nonlinear dynamical systems,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 51, 830–842 共2004兲. 7 J. Mason, P. S. Linsay, J. J. Collins, and L. M. Glass, “Evolving complex dynamics in electronic models of genetic networks,” Chaos 14, 707–715 共2004兲. 8 C. Li, X. Liao, and X. Yang, “Switch control for piecewise affine chaotic systems,” Chaos 16, 033104 共2006兲. 9 I. Cervantes, R. Femat, and J. Leyva-Ramos, “Study of a class of hydridtime systems,” Chaos, Solitons Fractals 32, 1081–1095 共2007兲. 10 G. F. V. Amaral, C. Letellier, and L. A. Aguirre, “Piecewise affine models of chaotic attractors: The Rössler and Lorenz systems,” Chaos 16, 013115 共2006兲. 11 E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130– 141 共1963兲. 12 C. T. Sparrow, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors 共Springer-Verlag, Berlin, 1982兲. 13 J. S. Birman and R. F. Williams, “Knotted periodic orbits in dynamical systems I: Lorenz’s equations,” Topology 22, 47–82 共1983兲. 14 C. Letellier and G. Gouesbet, “Topological characterization of reconstructed attractors modding out symmetries,” J. Phys. II 6, 1615–1638 共1996兲. 15 C. Letellier and L. Aguirre, “Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables,” Chaos 12, 549–558 共2002兲. 16 L. A. Aguire and C. Letellier, “Observability of multivariate differential embeddings,” J. Phys. A 38, 6311–6326 共2005兲. 17 C. Letellier and G. Gouesbet, “Topological characterization of a system with high-order symmetries: The proto-Lorenz system,” Phys. Rev. E 52, 4754–4761 共1995兲. 18 T. D. Tsankov and R. Gilmore, “Topological aspects of the structure of chaotic attractors in R3,” Phys. Rev. E 69, 056206 共2004兲. 19 R. Thomas, “Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘labyrinth chaos’,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, 1889–1905 共1999兲. 20 R. Thomas, “Multistationarity, the basis of cell differentiation and memory: I. Structural conditions of multistationarity and other nontrivial behavior,” Chaos 11, 170–179 共2001兲. 21 R. Thomas, On the Relation Between the Logical Structure of Systems and Their Ability to Generate Multiple Steady States or Sustained Oscillations, Springer Series in Synergetics Vol. 9 共Springer, Berlin, 1981兲, pp. 180– 193. 22 R. Thomas, “Logical description, analysis and synthesis of biological and other networks comprising feedback loops,” Adv. Chem. Phys. 55, 247– 282 共1984兲. 23 C. Letellier and O. Vallée, “Analytical results and feedback circuit analysis for simple chaotic flows,” J. Phys. A 36, 11229–11245 共2003兲. 24 C. Letellier, T. Tsankov, G. Byrne, and R. Gilmore, “Large-scale structural reorganization of strange attractors,” Phys. Rev. E 72, 026212 共2005兲. 25 G. Chen and T. Ueta, “Yet another chaotic attractor,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, 1465–1466 共1999兲. 26 Y. Wang, J. Singer, and H. H. Bau, “Controlling chaos in a thermal convection loop,” J. Fluid Mech. 237, 479–498 共1992兲. 27 T. Shimizu and N. Morioka, “On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model,” Phys. Lett. 76A, 201–204 共1980兲. 28 A. M. Rucklidge, “Chaos in models of double convection,” J. Fluid Mech. 237, 209–229 共1992兲. 29 R. Shaw, “Strange attractor, chaotic behavior and information flow,” Z. Naturforsch. A 36, 80–112 共1981兲. 30 J. S. Sprott, “Some simple chaotic flows,” Phys. Rev. E 50, R647–R650 共1994兲. 31 T. Rikitake, “Oscillations of a system of disk dynamos,” Proc. Cambridge Philos. Soc. 54, 89–105 共1958兲. 32 C. Letellier, P. Dutertre, J. Reizner, and G. Gouesbet, “Evolution of mul-

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp

023104-11

Chaos 17, 023104 共2007兲

Structure of Lorenz-like systems

timodal map induced by an equivariant vector field,” J. Phys. A 29, 5359– 5373 共1996兲. 33 J.-M. Malasoma, “What is the simplest dissipative chaotic jerk equation which is parity invariant?” Phys. Lett. A 264, 383–389 共2000兲. 34 J. C. Sprott and S. J. Linz, “Algebraically simple chaotic flows,” Int. J. Chaos Theory Applic. 5, 3–22 共2000兲. 35 C. Lainscsek 共private communication兲.

36

C. Lainscsek, C. Letellier, and C. Schürrer, “Ansatz library for global modeling with a structure selection,” Phys. Rev. E 64, 016206 共2001兲. 37 C. Letellier and E. A. M. Mendes, “Robust discretizations versus increase of the time step for the Lorenz system,” Chaos 15, 013110 共2005兲. 38 Z. Fu and J. Heidel, “Non-chaotic behavior in three-dimensional quadratic systems,” Nonlinearity 10, 1289 共1997兲.

Downloaded 01 Jun 2007 to 150.164.56.40. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp