Dimension Reduction of Dynamical Systems - Semantic Scholar

Nonlinear Dynamics (2005) 41: 1–15

c Springer 2005

Dimension Reduction of Dynamical Systems: Methods, Models, Applications GIUSEPPE REGA1,∗ and HANS TROGER2 1 Dipartimento

di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, via A. Gramsci 53, 00197 Rome, Italy; für Mechanik, Technische Universität Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria; ∗ Author for correspondence (e-mail: [email protected]) 2 Institut

(Received and accepted: 15 November 2004)

Abstract. After presenting some basic introductory ideas concerning dimension reduction and reduced order modelling, an overview of the contents of the papers collected in this Special Issue of Nonlinear Dynamics is given. Key words: approximate inertial manifold, center manifold, inertial manifold, nonlinear normal modes, proper orthogonal decomposition, reduced order models

1. Introduction To framework the papers collected in this Special Issue devoted to dimension reduction and reduced order modelling, we start with some general remarks and then shortly discuss the papers to give the interested reader a proper guidance. If an engineer looks into books on differential equations like those by Arnol’d [2], Guckenheimer and Holmes [8], Golubitsky, Schaeffer and Stewart [6], to name just a few, she or he might wonder why there is so much space devoted to low dimensional systems of ordinary differential equations since it is evident that accurate modelling of dynamic engineering processes and systems often requires high dimensional or even infinite dimensional models, resulting into systems of high dimensional ordinary differential equations or even partial differential equations. Hence an engineer typically is confronted in the description of the dynamics of the system with a representation space which is high dimensional, perhaps even of infinite dimension. The connection between these two apparently unconnected domains of dynamics is given by the methods of dimension reduction or reduced order modelling which currently is an active field of research both in mathematics and in engineering. In fact it is well known to engineers both from numerical simulations and from experiments, that for some systems, an accurate description of their asymptotic behaviour still should be possible by reducing the originally high dimensional space to a space of much smaller dimension. This heuristic observation is also evidenced by rigorous mathematics because it has been proved that certain dissipative partial differential equations possess finite dimensional inertial manifolds, which means that the long-term dynamics described by the partial differential equation is completely governed by a finite dimensional ordinary differential equation, without error. To demonstrate the idea behind reducing the order of a system, let us consider, for example, a system which performs a limit cycle oscillation. Even if this occurs for a system with many degrees of freedom, it should be possible to represent the dynamics of the full high dimensional system by the dynamics on a two dimensional phase space, because all different components of the system oscillate with the same

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G. Rega and H. Troger

limit cycle frequency but with different amplitudes, as it is well known from a Hopf bifurcation analysis via center manifold theory. In other words, if the investigated phenomenon is first presented in some arbitrary way, then, if the motion shows limit cycle behaviour, all dependent variables are represented as functions of just two distinguished variables. Such a behaviour can be well studied by the methods of local bifurcation theory which can be applied to the analysis of the nonlinear motion, setting in after loss of stability of an equilibrium. Center manifold theory, which is mathematically well founded and is more or less straight-forward to apply, will be shortly addressed below. Locally in the vicinity of a bifurcation point, where a considered state, be it an equilibrium or a periodic solution, loses stability, the center manifold attracts all solutions of the system and hence the nonlinear asymptotic dynamics of the full system is represented by the dynamics on the possibly very low dimensional center manifold. The main shortcoming of center manifold theory is that it is only a local theory, which means that, for example for the problem of loss of stability of an equilibrium at a critical parameter value, only small parameter variations about the critical value are allowed. However there exists a global equivalent to the center manifold, namely the inertial manifold, which is not restricted to the bifurcation scenario, just described, but which contains the whole asymptotic dynamics including all attractors of the system.

2. Inertial Manifolds and Approximate Inertial Manifolds The theory of inertial manifolds is more important from a theoretical point of view rather than from an application oriented point of view. The important message of the theory of inertial manifolds is that infinite dimensional systems may be precisely described in their long term behaviour by finite dimensional systems. However, from inertial manifold theory still very high dimensional systems may be obtained. This, from an application oriented point of view, still unpleasant fact is circumvented by the concept of approximate inertial manifolds or, in engineering language, nonlinear Galerkin methods. 2.1. INERTIAL MANIFOLDS In the global process of reducing an infinite dimensional system to a finite dimensional one, as usual, questions concerning existence and uniqueness of solutions, compactness of the universal attractor, estimation of the dimension of the attractor, and finally existence of a smooth finite dimensional invariant manifold called inertial manifold must be answered [7]. We only indicate the basic requirement for the existence of an inertial manifold which is a more delicate matter than the existence of universal attractors. The question that one asks is when a smooth invariant submanifold in a dynamical system will persist under perturbation. The attracting invariant manifold persistent under perturbations must have more extreme Lyapunov exponents in its normal directions than in its tangential directions. If the partial differential equations being studied have large gaps in their spectra, then these can be used to look for invariant manifolds that lie close to the linear space spanned by the modes whose eigenvalues lie to the right of a gap in the complex plane. To be more specific, let us consider a dissipative evolutionary equation of the form [9] u˙ = Au + F(u),

(1)

where A is a self-adjoint operator with compact resolvent, defined on a Hilbert space H and F(u) is the nonlinear part defined on the domain of A. Let u(t) = S(t)u 0 denote the solution to (1) at time t

Dimension Reduction of Dynamical Systems

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satisfying the initial condition u(t0 ) = u 0 . A is usually a (dissipative) spatial differential operator (like the Laplacian, or the biharmonic operator), making (1) a partial differential equation. The operator A has a complete orthogonal set of eigenfunctions w1 , w2 , . . ., with the real parts of the corresponding eigenvalues 0 > λ1 ≥ λ2 ≥ . . .. To see whether a gap exists let us consider for example the Laplacian A = in d-dimensional space. The eigenvalues λn are of the order λn ∼ −n 2/d and hence λn+1 − λn ∼ n

for d = 1

λn+1 − λn ∼ 1 for d = 2. In the gap condition [4] Lκ < 1

(2)

two quantities show up. One is the Lipschitz constant L, expressing that F is globally Lipschitz continuous with constant L and the other is κ which is inverse proportional to the length of the gap (for → ∞ we have κ → 0 [4]). Obviously for d = 1 condition (2) can be satisfied for large values of L. But for d = 2 increasing the number of modes, if the Lipschitz constant L is not small enough, does not result in fulfilling the inequality. We define P to be the spectral projection onto the span of the first n eigenfunctions and Q := I − P be that onto the remaining ones, i.e., Pu = p :=

n

α j (t)w j :=

j=1

n (u, w j ) wj (w j , w j ) j=1

(3)

and Qu = q :=

∞ j=n+1

α j (t)w j :=

∞ (u, w j ) wj (w j, wj) j=n+1

(4)

where (·, ·) denotes the scalar product in H . A subset M ⊂ H is said to be an Inertial Manifold for Equation (1) if (i) M is a finite-dimensional Lipschitz manifold in H , (ii) M is positively invariant, i.e., if u 0 ∈ M then S(t)u 0 ∈ M for all t > 0 and (iii) M is exponentially attracting, i.e., there is a ν > 0 such that for every u 0 ∈ H there is a constant K = K (u 0 ) such that dist(S(t)u 0 , M) ≤ K e−νt , t ≥ 0. The existence of an inertial manifold for a class of systems is very important since it implies that the long term dynamics of the original Equation (1) is completely described by a finite dimensional ordinary differential equation, without error. From the gap condition (2) it is possible to obtain an upper bound on n such that M is the graph of a smooth function : P H → Q H.

(5)

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Under the above assumptions the graph of the function is an n-dimensional manifold in H . For u ∈ H and using the projections (3) and (4), and the commutativity relations P A = A P and Q A = AQ we can write the partial differential equation (1) equivalently as the following system of ordinary differential equations p˙ = P Ap + P F( p + q),

(6)

q˙ = Q Aq + Q F( p + q).

(7)

By expressing the q variables in terms of the p variables through the relation q = ( p), we obtain the following reduced system of ordinary differential equations p˙ = P Ap + P F( p + ( p))

(8)

which can now be used to determine the long-term dynamics of the original equation without error. System (8) is called an inertial form of (1). Numerical methods [2] for solving (1), which in effect compute solutions of (8) with replaced by = 0, are called standard (linear, flat) Galerkin methods. Those, which use nontrivial approximations to the mapping in (8), are referred to as nonlinear Galerkin methods. Further note the way P and Q are defined in (3) and (4), P is an orthogonal projection on H with finite dimensional range, while Q has infinite dimensional range. 2.2. VARIOUS GALERKIN METHODS The main problem in the application of inertial manifold theory, provided an inertial manifold exists at all, to obtain a reduced order model is that usually the estimate of its dimension n is very high. In [10] it is shown that for the infinite dimensional model of a rotating beam the order of n is typically n ∼ 102 . Since such a high dimensional reduced order system is not useful for practical applications, various approximate methods called approximate inertial manifold theories or nonlinear Galerkin methods are used. These basically proceed in the following way. The infinite dimensional projection Q is approximated by truncating the series (4) by retaining only m modes and therefore Equation (4) is replaced by Qu = q :=

m j=n+1

α j (t)w j =

m (u, w j ) wj. (w j, wj) j=n+1

(9)

Because of this truncation, the relation between p and q is replaced by an approximation, say q = a ( p), which now maps P H into the finite-dimensional space Q H , and the corresponding graph Ma is now an approximate inertial manifold of (1), where in addition also n is chosen to be a small number. In [11] these approximate inertial manifold calculations are applied to the Kuramoto– Shivashinski equation, where it is shown that making use of the nonlinear Galerkin approach a much lower dimensional system can be obtained. Here a warning for engineers should be added. In engineering, in the derivation of low dimensional reduced systems, it often is assumed that the higher modes (the fast motion) in the representation of the system, can be completely ignored. However, due to nonlinear couplings one always should try to eliminate the fast motion (higher modes) in a way as it is done in the nonlinear Galerkin methods and not just ignore them. If the fast motion is completely ignored in the calculation of a low dimensional


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reduced system, which is equivalent to the standard Galerkin method, in many cases, if not sufficiently enough modes are retained, that is n is taken too small, this approach can fail badly and indicate wrong solution behaviour (see e.g. [11]). In some sense the representation of the higher modes by means of the lower modes is the essential step in applying more sophisticated dimension reduction methods. This elimination process of the fast (inessential, passive) modes is also called slaving of the fast modes to the slow modes. 2.3. KARHUNEN–LOEVE OR PROPER ORTHOGONAL DECOMPOSITION (POD) Another strong improvement of the Galerkin method is by application of the Karhunen–Loeve method. The Karhunen–Loeve method, which in the mathematical literature is also called Proper Orthogonal Decomposition (POD) method, has been quite successfully applied for the study of turbulence and coherent structures in fluid flow problems [17, 18] and more recently also in the characterisation of solid mechanics problems [1, 5]. However, its application requires some information about the dynamics of the considered system. From an ensemble of p data functions u i , obtained from experiments or by simulation, it generates a set of deterministic basis functions w j based on second-order statistics. These basis functions can be used in a standard Galerkin approximation. Basically an eigenvalue problem of the form (R(x, x ), w(x )) = R(x, x )w(x ) d x = λw(x). (10) D

has to be solved. The generalized function R(x, x ) = E{u(x)u(x )} is the covariance or the autocorrelation of u(x) and u(x ). It is assumed that u(x) is a regular function absolutely integrable on any finite region D. The solution of (10) supplies, first, the set of optimal eigenmodes wi and, second, the corresponding eigenvalues λi which can be interpreted as a measure of the energy contents carried by the corresponding mode. By optimal eigenfunctions or eigenvectors it is understood that they approximate the data in such a way that they are parallel to the axes of the inertia ellipsoid of the cloud of data points. It can be further shown that comparing two standard Galerkin reductions with the same number of ansatz functions, the POD modes give a more accurate result than any other set of ansatz functions. Moreover, by means of the normalized eigenvalues a hint on how many ansatz functions should be included to achieve a certain accuracy of the approximation can be obtained.

3. Center Manifold One of the main problems occuring in the approximate inertial manifold calculation, namely the determination of the dimension n, does not occur for the application of center manifold theory as we now shortly indicate. We assume that A = A(µ) in (1) depends on a parameter µ and consider the loss of stability of the trivial solution u = 0 of (1) under quasistatic variation of µ. For parameter values below µ = µc the solution u = 0 is supposed to be asymptotically stable. Under certain mild requirements [12] center manifold theory is applicable. Then the field variable u(x, t) is decomposed in the form u(x, t) = u c (x, t) + u s (x, t) =

n i=1

qi (t)wi (x) + U (qi (t), x),

(11)

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where the wi (x) are the active spatial modes, obtained from the solution of the eigenvalue problem related to the linear system u˙ = A(µc )u.

(12)

The qi (t) are their time dependent amplitudes and u s (x, t) could be given by an infinite sum. The decomposition is completely analogous to (3) and (4). The key difference in the application of center manifold theory compared to inertial manifold theory is that it is valid only locally. However the former has the advantage of providing an exact dimension of the reduced system instead of obtaining only an estimate of the dimension from (2). We assume that the spectrum of A(µ) is discrete and that for µ = µc a finite number (n) of eigenvalues crosses the imaginary axis at the same time. All other eigenvalues have a negative real part. Defining the projections as before, we obtain equations u˙ c = P Au c + P F(u c + u s ), u˙ s = Q Au s + Q F(u c + u s ),

(13)

formally completely analogous to (6) and (7). If u s = (u c ) is a smooth invariant manifold we call a center manifold if (0) = (0) = 0. Note that if in (13) P F = Q F = 0, all solutions tend exponentially fast to solutions of u˙ c = P Au c . That is, the linear n-dimensional equation on the (flat) center manifold determines the asymptotic behaviour of the entire infinite-dimensional linear system, up to exponentially decaying terms. The center manifold theorem [12] enables us to extend this argument to the nonlinear case, when P F and Q F are different from zero and to replace (13), if |u c | is sufficiently small, by u˙ c = P Au c + P F(u c + (u c )).

(14)

The zero solution of (13) has exactly the same stability properties as the zero solution of (14). Further for the determination of (u c ) the (partial) differential equation (u c )u˙ c = Q A(u c ) + Q F(u c + (u c ))

(15)

is obtained, from which a power series approximation of can be calculated. Summing up, the loss of stability is described in terms of the temporal evolution of the amplitudes of certain (active) modes, whose number and determination is clear for center manifold theory. These modes are those that are either mildly unstable or only slightly damped in linear theory. Their determination requires the solution of the linear eigenvalue problem (12). If the number of these critical modes is finite, a set of ordinary differential equations called the amplitude equations of the critical modes can be constructed, which govern the long term behaviour of the original system, at least locally in the neighborhood of the parameter value for which the loss of stability occurs. This dimension reduction scenario can be given a geometric interpretation in phase space (Figure 1). The evolution of the flow can be split into two components: The dynamics of the active modes, which dominate the long term behaviour, is governed by the nonlinear system (14), which also takes into account the rapidly decaying terms by the argument (u c ). If a solution of the full system starts on the invariant manifold, the passive modes are found directly from the equation u s = (u c ). For general initial values the solution converges quickly to the invariant manifold.


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Figure 1. Three-dimensional flow approching the two-dimensional center manifold on which a limit cycle represents the asymptotic behaviour after a Hopf bifurcation.

4. Slow Fast Dynamics in Conservative Systems From engineering experience it is also well known that in stiff systems by a proper choice of coordinates a separation in slow motions, which are dominant, and fast motions, which hardly affect the slow motions, may be performed. In other words the motions in the system are evolving on different time scales. One has a slowly evolving time scale describing the salient features of the system and a fast time scale, which is transient if the system is dissipative and oscillatory if the system is conservative. Simplification of the system dynamics concerning its dimension often can be achieved by elimination of the fast scales resulting in a reduction of the dimension of the original system. The so-called static condensation procedure often used to link fast variables to slow variables in certain continuous structural systems belongs to this class of procedures. A typical example are the oscillations of a string pendulum (Figure 2). If the string is stiff, negligible small fast axial oscillations will have almost no influence on the slow transversal and pendular oscillations of the whole system. If one introduces natural string coordinates which are elongation and the angle to the tangent vector [3] then the equations of motion [14] take the form of a singularly perturbed system. If one assumes that the essential dynamics occurs on a slow manifold then the slaving procedure mentioned before can also be applied for conservative systems. To show what type of questions have to be answered we consider as example the planar motion of a spring pendulum [13] consisting of a mass m and a very stiff spring (constant c, unstrained length l0 ). The equations of motion take the form p˙ = Aε (q) p + Fε (q, p) εq˙ = Bq + G ε (q, p)

(16) (17)

where p = (ϕ, ϕ), ˙

˙ q = (l, l).

(18)

Here ϕ designates the angle and l the elongation of the spring. The small parameter ωp ε= = ωs

mg l0 c

(19)

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Figure 2. Explanation of the fast and slow motions in a mathematically stiff system: left and middle frame show the slow transversal motions whereas in the right frame the fast axial motion is shown.

is the ratio of the small pendular frequency to the large extensional frequency. Hence p is the slow variable and q the fast variable. Now slaving the fast modes q by the slow modes p q = ε ( p)

(20)

similarly as it is done before, results in an equation of the form of (15) Bε + G ε (ε , p) = ε ε · [Aε (ε ) p + Fε (ε , p)]

(21)

from which an approximation of the slow manifold can be calculated, as long as the frequencies are not in resonance. This type of analysis has been done in [13] for the spring pendulum. On the resulting invariant manifold a two-dimensional motion in the four dimensional space is obtained. If the motion is exactly represented by the motion on the manifold, this motion is called in [15] a Nonlinear Normal Mode. Generalizing this concept we can say that a 2n-dimensional conservative system (n-degrees of freedom), the motion of which takes place on a two-dimensional invariant manifold, possesses a Nonlinear Normal Mode of motion. Hence the Nonlinear Normal Modes of a conservative system of the form (1) are synchronous oscillations of all components [16] taking place on a two dimensional invariant manifold. In the problem of the spring pendulum such a slow fast decomposition is possible. One can show that if one starts on the slow manifold the resulting oscillation is given by a Nonlinear Normal Mode. Another question, which is also raised in [13], namely the stability of the slow manifold motion, is an important but also difficult question. For systems of two degrees of freedom an answer is provided by KAM theory [18], whereas for higher dimensional systems more complicated mechanisms of instability may occur, such as Arnol’d diffusion.


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5. Contents of the Special Issue When planning this Special Issue on Dimension Reduction of Dynamical Systems, we thought it was important to give, as far as possible, an updated account of the several research topics which are being addressed within various scientific communities, and range from general theoretical methods to the development of reliably reduced models for different applications. As a matter of fact, the present issue collects 12 papers written by both applied mathematicians and engineers, which, according to the special preference of each author, represent either possibly tutorial articles on methods, or articles – both overviewing and presenting new results – on applications and reduced-order models. On the one hand, various approaches to dimension reduction including POD, Spectral Decomposition, Nonlinear Normal Mode Theory, Multiple Time Scaling, Galerkin Discretization, and Time Series Analysis, are addressed, while on the other hand, implementation of methods and development of reduced models for specific applications are highlighted. Overall, the style of the papers dealing with fundamentals appears accessible to theoretically oriented engineers, assuming them not to be necessarily expert in the relevant field. In turn, papers concerned with applications do report on updated issues and technical problems encountered in a variety of engineering or physical systems. As reflected in the title of the Special Issue, it seems worth frameworking all collected papers into two main subtopics identifiable within the general issue topic, namely Methods for Dimension Reduction, and Applications and Reduced-Order Models, though, of course, they somehow overlap with each other. Accordingly, in the sequel, we shortly report on the specific contents of each paper by merging it into one – or both – of the above mentioned subtopics, along with the contents from companion papers. 5.1. METHODS FOR DIMENSION REDUCTION Though being also involved, to a different extent, with modelling and application aspects, a few papers basically deal with the method of POD (also called Karhunen–Loeve decomposition or principal components analysis or singular value decomposition) and give overviews of the relevant theory: • G. Kerschen, J.C. Golinval, A.F. Vakakis, and L.A. Bergman in “The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical System: An Overview”, updatedly overview the general features of the POD method, by recalling its mathematical formulation, by explaining the different means of computing the decomposition, and by providing some insight into the physical interpretation of the modes extracted from the POD. This includes the relationship between proper orthogonal modes (POMs) and linear mode shapes or nonlinear normal modes (NNMs). The POD method is presented as an interesting alternative to modal analysis and NNM theory for the dynamic characterization and order reduction of linear and nonlinear mechanical systems. Moreover, its strengths and limitations are discussed by also referring to some new statistical techniques (nonlinear principal component analysis, independent component analysis) capable to overcome the existing weaknesses of the POD method in accounting for the nonlinear correlations between variables. • P. Glösmann and E. Kreuzer in “Nonlinear System Analysis with Karhunen–Loeve Transform” also give an overview of the KLD and of its current applications. After shortly summarizing the relevant mathematical concepts, the authors comprehensively highlight those adaptive features of the theory which entitle it to be used as a universal tool for various tasks, which include signal analysis, model reduction, optimal control, template matching, and compressive image encoding. For the purpose of dynamical systems analysis, the KLD has been applied to stationary states, only. The authors show how it also helps to investigate nonlinear dynamical systems during states of non-stationary

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behaviour, by focusing on approaches to the relevant state monitoring based on experimental data. POD is dealt with as a proper tool to investigate on low-dimensional modelling of turbulence in the tutorial paper by • T.R. Smith, J. Moehlis and P. Holmes: “Low-dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial”, in which, besides providing a general overview of the procedure, the authors describe two different ways to numerically calculate the POD modes, discuss how one can exploit symmetry considerations to simplify and understand them, comment on how parameter variations are captured naturally in ODE models. In turn, • I. Georgiou in “Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Oscillation and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods” also makes use of the POD method, but he mainly aims at developing suitable reduced order models to explore dimensional complexity of the dynamics of a coupled multi-field structural system. Three papers deal with the general topic of projection methods, somehow addressing more fundamental aspects. • D.S. Broomhead and M. Kirby in “Dimensionality Reduction using Secant-based Projection Methods: The Induced Dynamics in Projected Systems” deal with the (large) class of semi-analytical, or empirical–analytical, methods for reduction of dynamical systems, which can be considered to include the optimal linear POD, too. The approach to the data reduction problem taken in the authors’ investigation is based on Whitney’s embedding theorem and is the result of the evolution of many algorithms for computing reduced order models. It involves picking projections of the high-dimensional system which are optimised in the sense that they are easy to invert. The paper investigates whether the existence of an easily invertible projection leads to practical methods for the construction of an equivalent, low-dimensional dynamical system. After reviewing the secant-based projection method and simple methods for finding good representations of the nonlinear inverse of the projections, two variants of a way to find the induced dynamical system are discussed, along with the related different numerical approximations. One novel aspect of the approach is in describing various ways in which knowledge of the full dynamical system can be incorporated into the reduced system. The effects of the procedure are demonstrated on a simple system of nonlinear ODEs exhibiting an attracting limit cycle, and on low-dimensional solutions of the Kuramoto–Sivashinsky equation which need many Galerkin modes for their description. • S.C. Sinha, S. Redkar, V. Deshmukh and E.A. Butcher in “Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications” give a comprehensive overview of order reduction techniques for nonlinear systems with time periodic coefficients, which occur in several technical problems, by considering equations of motion formulated in either state-space or secondorder (structural) form. Four techniques are illustrated in the former case, all of them making use of the Lyapunov–Floquet transformation to obtain equivalent systems with time invariant linear parts and time periodic nonlinear parts. The non-dominant dynamics are neglected in the linear technique, whereas they are accounted for, to a different extent, in the other techniques, namely a nonlinear projection method based on singular perturbation, a post-processing method, and the invariant manifold technique. The latter assumes a nonlinear time periodic relationship between dominant (master) and non-dominant (slave) states, as in the time periodic center manifold theory, and yields a reducibility condition providing conditions under which an accurate nonlinear order reduction is possible in the presence of strong parametric excitation. When deriving reduced order models in direct second-order


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form, L.-F. and other canonical transformations are applied, a master–slave separation of degreesof-freedom is performed, and a relevant nonlinear relationship is constructed. The same reducibility conditions as those given by the invariant manifold approach in state-space, are obtained. Application and comparison of various techniques is presented, and several kinds of resonance conditions are derived. The paper ends by discussing the straightforward generalization of state-space techniques to a special class of periodic–quasiperiodic systems, apart from the case of the invariant manifold approach where a nonlinear quasiperiodic relationship between non-dominant and dominant states has to be assumed. Two main motivations stand in the background of the theoretical paper by • I. Mezic: “Spectral Properties of Dynamical Systems, Model Reduction and Decompositions”, namely: (i) the interest in improving projection methods for obtaining low-dimensional models of infinite-dimensional systems by introducing stochastic terms to account for neglected modes; (ii) the observation that an analysis of how the dynamics on the attractor of a system that is being reduced, affects the reduction itself is seldom found in model reduction approaches. The author discusses two issues related to model reduction of deterministic or stochastic processes, that are directly related to the asymptotic properties of the dynamics. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional system with the properties and possibilities for model reduction. After reviewing some elements of the spectral theory of dynamical systems, this is applied to obtain a new type of decomposition, that combines spectral decomposition to extract the almost periodic part of the evolving process, and POD decomposition to the remainder of the process which has continuous spectrum. The author speculates that finite dimensional truncations should do well in this case. Along this line, the second addressed topic is model validation, where the original, high-dimensional dynamics and the dynamics of the reduced model are compared in some norm. It is argued that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection. Nonlinear normal mode theory and multiple time scale analysis are respectively considered in two more papers dealing with methods for dimension reduction. • P. Apiwattanalunggarn, S.W. Shaw, and C. Pierre in “Component Mode Synthesis Using Nonlinear Normal Modes” describe an interesting methodology for developing reduced-order dynamic models of structural systems composed of an assembly of nonlinear component structures, possibly modelled by finite-element techniques. The approach is a nonlinear extension of the fixed-interface component mode synthesis (CMS) technique previously developed for linear structures, and combines formulation of the NNM invariant manifolds of the individual substructures – which allows one to obtain relevant single-mode nonlinear representations – with the fixed-interface nonlinear CMS. The general idea is that of accounting for nonlinearities at the substructure level, without having to resort to the retention of several linear modes as in the original fixed-interface linear CMS. • A. Luongo and A. Di Egidio in “Bifurcation Equations through Multiple-Scales Analysis for a Continuous Model of a Planar Beam” deal with reduction methods in nonlinear dynamics, and attempt to extend to infinite-dimensional systems a multiple-scale approach to bifurcation equations previously developed by the authors for finite-dimensional systems. The method makes it possible to avoid both the search for the center manifold and the use of the normal form theory, since the algorithm furnishes the bifurcation equations governing the system asymptotic dynamics directly in normal form. The key point of the authors’ engineering-based approach lies in assuming the possibility to extend to differential operators the well-known properties of the Jordan chain of generalized eigenvectors of algebraic operators, which entails stating the differential equations of infinite-dimensional systems

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G. Rega and H. Troger in an operator form, in order to be dealt with as if they were algebraic equations. The various steps of the method are highlighted with reference to a continuous model of internally constrained, planar beam, equipped with a visco-elastic device and loaded by a follower force. Codimension-1 and -2 bifurcations are analysed, and closed-form solutions are obtained for each of the three forms of instability (divergence, Hopf, double-zero bifurcation) of the trivial equilibrium of the system. A numerical approach will likely be necessary for more involved equations.

5.2. APPLICATIONS AND REDUCED-ORDER MODELS We start this subsection by noticing how a number of interesting applications of POD to nonlinear dynamical systems in solid and structural mechanics are included in this Special Issue, thus meaningfully widening the area of technical interest of a method originally applied to systems in fluid mechanics. Accordingly, we briefly report on applications of dimension reduction methods, and on the ensuing reduced order modelling, starting just from POD applications to solid mechanics. Also worth of interest is the observation that the dimension reduction effort in the various papers is mostly devoted towards obtaining suitable reduced order models of actually infinite-dimensional (or continuous) systems as occurring in either nonlinear solid or fluid dynamics. In the following, both papers already mentioned in the previous subsection and new papers are referred to. A literature survey of applications of POD is presented in the paper by Kerschen et al., with a special emphasis on the applications in the field of structural dynamics, which are becoming increasingly popular in the last few years. POD analysis is used to study nonlinear dynamical effects in continuous systems due to vibro-impacts, and is shown to be an effective non-parametric system identification tool that can be used for diagnosis and monitoring of the performance of vibrating structural assemblies. The authors also discuss the capability of the POD method to effectively reduce the order of the dynamics of spatially extended continuous systems with high modal densities – which are typical in certain civil engineering and aerospace applications – where traditional modal analysis methods are difficult to apply. In particular, POM-based reduced-order models of dimensions m equal or less than 6 are shown to effectively reproduce the linear dynamics of a truss structure. Glösmann and Kreuzer present a new application of experimental-based KLD to non-linear dynamical systems in non-stationary conditions. The investigation is aimed at monitoring the state of a poorly observable and highly nonlinear dynamical system, namely a rolling railway wheelset, and at observing the relevant changes. The KLD is applied to measurement data gathered from a railway wheelset experiment. Based also on the investigation of the sensitivity of the analysis to changes in signal properties, the usefulness of KLD in the state monitoring of non-stationary dynamics is shown to consist mostly in its capability to make state changes more evident. The tutorial paper by Smith et al. on low-dimensional modelling of turbulence, recovers the original interest of the fluid dynamics community towards POD applications. Plane Couette flow (PCF) is considered as a simple, instructive, fluid system for the application of POD techniques to a minimal flow unit, namely a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence. The authors discuss how the discrete symmetries of PCF can be used to simplify and understand POD modes – herein derived from a direct numerical simulation database – and show how, with suitable modelling of neglected modes, very low-dimensional models can be able to capture many aspects of turbulent fluid flows. In particular, how one can uncouple POD modes to allow streamwise and crossstream components to evolve independently, thus removing non-physical constraints imposed by very low-dimensional truncations. Then, two different low-dimensional models for turbulence are discussed, with a six-mode model involving coupled POD modes, and a nine-mode model using uncoupled modes,


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by also emphasizing that the best model is not necessarily obtained when keeping the most energetic modes, and by mentioning a few extensions of the POD which may lead to improved models for some fluid systems. A couple of papers are concerned with developing suitable reduced order models of structural systems based on the relevant normal modes of vibration. To illustrate the applicability and explore the accuracy of the proposed NNM-based Component Mode Synthesis technique, Apiwattanalunggarn et al. consider a class of nonlinear spring-mass systems with weak coupling, and make extended comparisons of the performance of the reduced-order (threeDOF) nonlinear-CMS model with those of a number of other models, ranging from the full nonlinear original one (forty-one-DOF) to full/reduced models with different approximations. Simulation results show that three-DOF models (both linear and nonlinear) not accounting for contributions from higher fixed-interface linear modes show significant errors, whereas the three-DOF nonlinear-CMS model outperforms the three-DOF one obtained from the classical fixed-interface linear-CMS approach, though having just limited computational advantages with respect to the full nonlinear model accounting for all fixed-interface linear modes. • P.B. Goncalves and Z.J.G.N. del Prado in “Low Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells” also address reduced order modelling in nonlinear structural dynamics. Relying on a perturbation procedure developed in previous works to identify the essential nonlinear vibration modes of a thin-walled shallow cylindrical shell, the authors discuss the derivation of low-dimensional Galerkin models for the analysis of the relevant finite-amplitude vibrations under harmonic axial excitation. The responses of several reduced-order models are compared with each other, by analysing the influence of low-order modelling on the convergence of critical loads, bifurcation diagrams, attractors and large amplitude responses. It is shown that rather low-dimensional models, properly selected to account for the inherent in-out asymmetry of the displacement field of the shell surface, can describe with good accuracy the relevant response up to very large vibration amplitudes. The influence of the nonlinear modal interaction due to the shell’s high modal density on system dynamics is also discussed, showing the importance of a correct choice of the minimal retained number of modes to the aim of catching the correct shell behaviour. The paper by Georgiou can somehow be considered to establish a link between the two already addressed issues of POD modes and normal modes of vibration. It deals with the POD processing of the finite element dynamics of a geometrically exact nonlinear rod, and aims at deriving reduced order models capable to faithfully reproduce the dynamics of the full-order system in a wide range of forcing parameters, as well as to search for unknown relations between POD modes and normal modes of vibration. The author derives three-DOF reduced models through direct Galerkin projections of the system coupled equations of motion onto POD modes, and uses them to pave the way towards a normal mode characterization of coupled vibrations. As a matter of fact, the basic issue in the paper consists just in highlighting when POD modes represent normal modes of vibration. Categories of motion dominated by a single POD mode – but actually characterized by three DOF representing both bending-shearing and longitudinal motion of the rod, according to a master–slave relationship – are investigated in-depth. Based on the eigenvalue structure of the reduced system and numerical investigations, clear hints are obtained that the reduced order model possesses a global slow invariant manifold representing the signature for a non-synchronous normal mode of vibration. Overall, the POD-based reduced model method shows to be a potentially valuable tool to explore the spatio-temporal complexity of the dynamics of coupled multi-field structural systems. Specific application fields are considered in two more papers dealing with reduced-order models.

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• A.H. Nayfeh, M.I. Younis, and E.M. Abdel-Rahman in “Reduced-Order Models for MEMS Applications” updatedly review the development of reduced-order models for MEMS devices, which have registered a considerable deal of recent research activity as a suitable way to balance the need for enough fidelity in the model against the numerical efficiency needed for its practical use. Based on their implementation procedures, reduced-order models are classified into two broad categories: node and domain methods. As local approximations of the original PDEs, obtained by evaluating them at each node in the discretization mesh, node methods perform rather poorly in predicting transients, large motions, and nonlinear behaviour. In turn, domain-based methods eliminate the spatial dependence through the Galerkin method by either extracting a proper basis set from finite-element/finite-difference/experimental time series or employing the mode shapes of the MEMS device. The authors overview a number of reduced-order models for electrically actuated microbeams and rectangular/circular microplates they have recently developed through the latter domain method. They are validated with available theoretical and experimental results. Moreover, they present reduced-order approaches to model squeeze-film and thermoelastic damping in MEMS, based on solving analytically the equations governing the energy dissipation and extracting explicit analytical expressions for the damping coefficients. The aim is to decouple the coupled fluidic, thermal, and structural domains of MEMS devices, and to suitably reduce the computational cost of the relevant simulations. Finally, the basically numerical paper by • I. M. Moroz: “The Extended Malkus–Robbins Dynamo as a Perturbed Lorenz System”, deals with the nonlinear dynamics of one particular model of self-exciting Faraday-disk dynamos. Besides being nonlinear electro-mechanical engineering devices, the importance of such dynamos rests in their containing some of the key ingredients of large-scale naturally-occurring magneto-hydrodynamic fluid systems, while being of considerably lower dimension and so much more amenable to systematic investigations. Rich ranges of irregular solutions are exhibited by different low-order dynamo models in this family, which can be distinguished via the identification of the spectrum of unstable periodic orbits (upos) and the bifurcation sequences under variations of key parameters. This is achieved in the present study for the extended Malkus–Robbins dynamo, which is a set of four coupled nonlinear ODEs having the added interest of reducing to the Lorenz equations when one of the key bifurcation parameters vanishes. Since the structure of the upos of these equations has received considerable attention in the literature, perturbations away from the Lorenz limit are considered as an ideal starting point to characterize modifications of system response: a numerical investigation is accomplished of what befalls lowest order unstable periodic orbits – which contain many properties of the system chaotic time series – as well as return maps.

6. Concluding Remarks It is worth ending this introductory overview on dimension reduction of dynamical systems with few final points. 1. Besides the well established methods for dimension reduction described in Sections 2–4, further new projection techniques are currently being developed, mostly with the aim of overcoming some weaknesses of existing methods in properly accounting for system nonlinearities. 2. A variety of new applications of diverse techniques to different dynamical systems are continuously appearing in the scientific literature, meaningfully enriching the basic knowledge in the relevant field and clarifying the likely existing relations between independently developed procedures.


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3. Reduced order modelling is a very important issue for technical applications, as regards both finitedimensional and, mostly, infinite-dimensional systems occurring in either science or engineering. Previous points are also partially reflected in the overall contents of this issue, which we hope will be found interesting and useful by the readers of Nonlinear Dynamics dealing with such problems in their scientific activity. Finally, we express our sincere thanks to all of the authors for accepting to contribute to this issue, as well to the Journal Editor for hosting it.

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