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Strange Attractors and Classical Stability Theory ST. PETERSBURG STATE UNIVERSITY G. A. LEONOV STRANGE ATTRACTORS AND CLASSICAL STABILITY THEORY ST. PETERSBURG UNIVERSITY PRESS Published by St. Petersburg University Press 11/21 6th Line, St. Petersburg, 199004, Russia Leonov G. A. Strange Attractors and Classical Stability Theory ISBN 978-5-288-04500-4 Copyright c 2008 by St. Petersburg University Press 1 Подписано в печать 18.12.2007. Формат 60×90 /16. Бумага офсетная. Печать офсетная. Усл. печ. л. 10. Тираж 500 экз. Заказ № Типография Издательства СПбГУ. 199061, С.-Петербург, Средний пр., 41 Printed in Russia by St. Petersburg University Press Contents Preface 7 1. Definitions of Attractors 11 2. Strange Attractors and the Classical Definitions of Instability 21 2.1 Basic Definitions in Classical Stability of Motion . 21 2.2 Relationships Between the Basic Stability Notions . 24 2.3 Trajectory Sensitivity to Initial Data and the Basic Notions of Instability . 28 2.4 Reduction to a Study of the Zero Solution . 33 3. Characteristic Exponents and Lyapunov Exponents 35 3.1 CharacteristicExponents.................. 35 3.2 Lyapunov Exponents . 37 3.3 NormEstimatesfortheCauchyMatrix.......... 38 3.4 The Counterexample of Nemytskii—Vinograd [20] . 41 4. Perron Effects 43 5. Matrix Lyapunov Equation 49 6. Stability Criteria by the First Approximation 55 7. Instability Criteria 61 7.1 The Perron—Vinograd Triangulation Method . 62 7.2 Theorems on Instability . 65 7.3 Conclusion.......................... 75 5 6 Strange Attractors and Classical Stability Theory 8. Stability in the Large 77 9. Zhukovsky Stability 81 10. Lyapunov Functions in the Estimates of Attrac- tor Dimension 95 11. Homoclinic Bifurcation 113 12. Frequency Criterion for Weak Exponential In- stability on Attractors of Discrete Systems 129 12.1 The Yakubovich—Kalman and Kalman—SzegoLemmas.................. 130 12.2 Definitions of Curve Length and Weak Exponential Insta- bility . 134 12.3 Frequency Instability Criterion . 138 13. Estimates of Oscillation Period in Nonlinear Discrete Systems 145 References 151 Index 159 Preface This book might be more accurately titled, “On the advantages of the classical theory of stability of motion in the study of strange attractors.” In almost any solid survey or book on chaotic dynamics, one encoun- ters notions from classical stability theory such as Lyapunov exponent and characteristic exponent. But the effect of sign inversion in the character- istic exponent during linearization is seldom mentioned. This effect was discovered by Oscar Perron [110], an outstanding German mathematician, in 1930. The present book sets forth Perron’s results and their further de- velopment (see [79, 80, 81]). It is shown that Perron effects may occur on the boundaries of a flow of solutions that is stable by the first approxima- tion. Inside a flow, stability is completely determined by the negativeness of the characteristic exponents of linearized systems. It is often said that the defining property of strange attractors is the sensitivity of their trajectories with respect to the initial data. But how is this property connected with the classical notions of instability? Many researchers suppose that sensitivity with respect to initial data is adequate for Lyapunov instability [3, 34]. But this holds only for discrete dynamical systems. For continuous systems, it was necessary to remember the almost forgotten notion of Zhukovsky instability. Nikolai Egorovich Zhukovsky, one of the founders of modern aerodynamics and a prominent Russian scientist, introduced his notion of stability of motion in 1882 (see [142, 143]) — ten years before the publication of Lyapunov’s investigations (1892 [94]). The notion of Zhukovsky instability is adequate to the sensitivity of trajecto- ries with respect to the initial data for continuous dynamical systems. In this book we consider the notions of instability according to Zhukovsky [142], Poincar´e [117], and Lyapunov [94], along with their adequacy to the sensitivity of trajectories on strange attractors with respect to the initial data. 7 8 Strange Attractors and Classical Stability Theory In order to investigate Zhukovsky stability, a new research tool — a moving Poincar´e section — is introduced. With the help of this tool, exten- sions of the widely-known theorems of Andronov—Witt, Demidovic, Borg, and Poincar´e are carried out. At the present time, the problem of justifying nonstationary lineariza- tions for complicated, nonperiodic motions on strange attractors bears a striking resemblance to the situation that occurred 120 years ago. J.C. Maxwell (1868 [97, 98]) and I.A. Vyshnegradskii (1876 [98, 135]), the founders of automatic control theory, courageously used linearization in a neighborhood of stationary motions, leaving the justification of such linearization to H. Poincar´e (1881–1886 [117]) and A.M. Lyapunov (1892 [94]). Now many specialists in chaotic dynamics believe that the positive- ness of the leading characteristic exponent of a linear system of the first approximation implies the instability of solutions of the original system (e.g., [12, 47, 52, 103, 105, 118]). Moreover, there is a great number of computer experiments in which various numerical methods for calculating characteristic exponents and Lyapunov exponents of linear systems of the first approximation are used. As a rule, authors largely ignore the jus- tification of the linearization procedure and use the numerical values of exponents thus obtained to construct various numerical characteristics of attractors of the original nonlinear systems (Lyapunov dimensions, metric entropies, and so on). Sometimes computer experiments serve as arguments for partial justification of the linearization procedure. For example, com- puter experiments in [119] and [105] show the coincidence of the Lyapunov and Hausdorff dimensions of the attractors of Henon, Kaplan—Yorke, and Zaslavskii. But for B-attractors of Henon and Lorenz, such coincidence does not hold (see [75, 78]). So linearizations along trajectories on strange attractors require justifi- cation. This problem gives great impetus to the development of the nonsta- tionary theory of instability by the first approximation. The present book describes the contemporary state of the art of the problem of justifying nonstationary linearizations. The method of Lyapunov functions — Lyapunov’s so-called direct method — is an efficient research device in classical stability theory. It turns out that even in the dimension theory of strange attractors one can progress by developing analogs of this method. This interesting line of investigation is also discussed in the present book. When the parameters of a dynamical system are varied, the structure of its minimal global attractor can change as well. Such changes are the Preface 9 subject of bifurcation theory. Here we describe one of these, namely the homoclinic bifurcation. The first important results concerning homoclinic bifurcation in dissipa- tive systems was obtained in 1933 by the outstanding Italian mathematician Francesco Tricomi [133]. Here we give Tricomi’s results along with similar theorems for the Lorenz system. For the Lorenz system, necessary and sufficient conditions for the exis- tence of homoclinic trajectories are obtained. Lyapunov’s direct method, when combined with the efficient algebraic apparatus of modern control theory, is highly efficient for the study of chaos in discrete dynamical systems. It is usual for discrete systems to have no pictorial representations in phase space, which are characteristic of contin- uous systems with smooth trajectories that fill this space. However, this drawback can be partially overcome if, instead of observing changes in the distance between two solutions in a discrete time, one considers the line segment joining the initial data at the initial moment and its iterations — a sequence of continuous curves. A change in the length of these curves provides information on the stability or instability and other properties of solutions of discrete systems. The consideration of the lengths of such continuous curves may serve as a basis for definitions of stability and insta- bility, which is of particular importance for noninjective maps determining respective discrete systems (at present, it is precisely such maps that are widely-known generators of chaos). The matter is that in some cases two trajectories may “stick together” for a finite number of iterations. But the iterated segment that joins the initial data may have an increasing length even with regard to such sticking together. It turned out that in a number of cases, the mechanism of stretching the lengths is rather strong, and it can be revealed and estimated with the help of the whole rich arsenal of Lyapunov’s direct method, which can be combined with an efficient alge- braic apparatus: the Kalman—Szeg¨o and Shepelyavyi lemmas. These yield estimates of the stretching of the lengths of iterated curves in terms of fre- quency inequalities efficiently verifiable for particular dynamical systems. The methods mentioned above are described in detail in the present book. For the last fifty years another interesting direction in stability the- ory has been developed — the method of aprioriintegral estimates, which is based on the application of Fourier transforms as unitary operators in certain functional spaces. In the book it is shown that this method may pro- vide informative estimates for the oscillation periods of discrete dynamical systems. Its application to different one-dimensional maps yields estimates 10 Strange Attractors
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