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Ergodic Theory of Chaos and Strange Attractors J.-P Ergodic theory of chaos and strange attractors J.-p. Eckmann Universite de Geneve, 1211 Geneve 4, Switzerland D. Ruelle Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are pro- vided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quanti- ties, as well as their experimental determination, are discussed. The systematic investigation of these quan- tities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids. CONTENTS IV Entropy and Information Dimension 637 I. Introduction 617 Entropy 637 II. Differentiable Dynamics and the Reconstruction of B. SRB measures 639 Dynamics from an Experimental Signal 621 C. Information dimension 641 A. What is a differentiable dynamical system? 621 D. Partial dimensions 642 B. Dissipation and attracting sets 622 E. Escape from almost attractors 643 C. Attractors 623 F. Topological entropy* 644 D. Strange attractors 624 G. Dimension of attractors 644 E. Invariant probability measures 625 H. Attractors and small stochastic perturbations* 644 F. Physical measures 626 1. Small stochastic perturbations 644 G. Reconstruction of the dynamics from an experimental 2. A mathematical definition of attractors 645 signal 627 I. Systems with singularities and systems depending on H. Poincare sections 627 time 646 I. Power spectra 628 V, Experimental Aspects 646 J. Hausdorff dimension and related concepts 629 A. Dimension 646 III. Characteristic Exponents 629 1. -Remarks on physical interpretation 647 A. The multiplicative ergodic theorem of Oseledec 629 a. The meaningful range for C (r) 647 B. Characteristic exponents for differentiable dynamical b. Curves with "knees" 647 systems 630 c. Spatially localized degrees of freedom 648 1. Discrete-time dynamical systems on IR 630 2. Other dimension measurements 649 2. Continuous-time dynamical systems on R 630 B. Entropy 649 3. Dynamical systems in Hilbert space 631 C. Characteristic exponents: computer experiments 650 4. Dynamical systems on a manifold M 631 D. Characteristic exponents: physical experiments 651 C. Steady, periodic, and quasiperiodic motions 631 E. Spectrum, rotation numbers 652 1. Examples and parameter dependence 631 VI. Outlook 653 2. Characteristic exponents as indicators of periodic Acknowledgments 653 motion 632 References 653 D. General remarks on characteristic exponents 632 1. The growth of volume elements 632 2. Lack of explicit expressions, lack of continuity 632 3. Time reflection 633 4. Relations between continuous-time and discrete- time dynamical systems 634 i. iNTRODuCT~OV 5. Hamiltonian systems 634 E. Stable and unstable manifolds 634 In recent years, the ideas of differentiable dynamics F. Axiom- A dynamical systems 636 1. Diffeomorphisms 636 have considerably improved our understanding of irregu- 2. Flows 636 lar behavior of physical, chemical, and other natural phe- 3. Properties of Axiom- A dynamical systems 637 nomena. In particular, these ideas have helped us to G. Pesin theory* 637 understand the onset of turbu1ence in fluid mechanics. There is now ample experimental and theoretical evidence that the qua1itative features of the time evolution of many *Sections marked with + contain supplementary material physical systems are the same as those of the solution of a which can be omitted at first reading. typical evolution equation of the form Reviews of Modern Physics, Vol. 57, No. 3, Part I, July 1985 Copyright 1985 The American Physical Society 617 618 J.-P. Eckmann and O. Ruelle: Ergodic theory of chaos x(t) =F„{x(t)), x E R modes. Each mode is periodic, and its state is represented by an angular variable. The global system is quasiperiodic in a space of small dimension m. Here, x is a set of coor- (i.e., a superposition of periodic motions). From this per- dinates describing the system mode amplitudes, (typically, spective, a dissipative system becomes more and more tur- concentrations, etc. and determines the nonlinear ), F& bulent as the number of excited modes grows, that is, as time evolution these modes. The corre- of subscript p the number of independent oscillators needed to describe an experimental control parameter, which is sponds to the system progressively increases. This point of view is kept constant in each run of the experiment. (Typically, very widespread; it has been extremely useful in physics is the intensity of the force driving the system. } We p and can be formulated quite coherently (see, for example, write Haken, 1983). However, this philosophy and the corre- x(t)=f„'(x(0)) . (1.2) sponding intuition about the use of Fourier modes have to be completely modified when nonlinearities are impor- We usually assume that there is a parameter value, say tant: even a finite dimen-sional motion need not be quasi p=0, for which the equation is well understood and leads periodic in genera/. In particular, the concept of "number to a motion in phase space which, after some transients, of excited modes" will have to be replaced by new con- settles down to be stationary or periodic. cepts, such as "number of non-negative characteristic ex- As the parameter p is varied, the nature of the asymp- ponents" or information dimension. " These new con- ' totic motion may change. The values p for which this cepts come from a statistical analysis of the motion and change of asymptotic regime happens are called bifurca- will be discussed in detail below. tion points. As the parameter increases through succes- In order to talk about a statistical theory, one needs to sive bifurcations, the asymptotic motion of the system say what is being averaged and in which sample space the typically gets more complicated. For special sequences of measurements are being made. The theory we are about these bifurcations a lot is known, and even quantitative to describe treats time averages. This implies and has the features are predicted, as in the case of the period- advantage that transients become irrelevant. (Of course, doubling cascades ("Feigenbaum scenario"). We do not, there may be formidable experimental problems if the however, possess a complete classification of the possible transients become too long. ) Once transients are over, the transitions to more complicated behavior, leading eventu- motion of the solution x of Eq. (1.1) settles typically near ally to turbulence. Geometrically, the asymptotic motion a subset of R, called an attractor (mathematical defini- follows an attractor in phase space, which will become tions will be given later). In particular, in the case of dis- more and more complicated as p increases. sipative systems, on which we focus our attention, the review is The aim of the present to describe the current volume occupied by the attractor is in general very small state of the theory of statistical properties of dynamical relative to the volume of phase space. We shall not talk systems. This theory becomes relevant as soon as the sys- about attractors for conservative systems, where the "excited" tem is beyond the simplest bifurcations, so that volume in phase space is conserved. For dissipative sys- at- precise geometrical information about the shape of the tems we may assume that phase-space volumes are con- the motion on it is no available. See tractor or longer tracted by the time evolution (if phase space is finite di- Eckmann (1981) for a review of the geometrical aspects of mensional). Even if a system contracts volumes, this does dynamica1 systems. The statistical theory is still capable not mean that it contracts lengths in all directions. Intui- at- of distinguishing different degrees of complexity of tively, some directions may be stretched, provided some tractors and motions, and presents thus a further step in others are so much contracted that the final vo1ume is bridging the gap between simple systems and fully smaller than the initial volume (Fig. 1). This seemingly developed turbulence. In particular, the present treatment trivial remark has profound consequences. It implies does not exclude the description of space-time patterns. that, even in a dissipative system, the final motion may be After introducing precise dynamical concepts in Sec. II, unstable within the attractor. This instability usually in we address the theory of characteristic exponents Sec. manifests itself by an exponential separation of orbits (as III and the theory of entropy and information dimension time goes on) of points which initially are very close to in Sec. IV. In Sec. V we discuss the extraction of dynami- each other (on the attractor). The exponential separation cal quantities from experimental time series. takes place in the direction of stretching, and an attractor It is necessary at this point to clarify the role of the having this stretching property will be called a strange at- physical concept of mode, which appears naturally in sim- tractor. We shall also say that a system with a strange at- ple theories (for instance, Hamiltonian theories with tractor is chaotI'c or has sensitive dependence on initial quadratic Hamiltonians), but which loses its importance conditions. Of course, since the attractor is in general in nonlinear dynamical systems. The usual idea is to bounded, exponential separation can only hold as long as represent a physical system by an appropriate change of distances are small.
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