Nonlinear Dynamics of Chaotic and Stochastic Systems

We present an improved and enlarged version of our book Nonlinear - namics of Chaotic and Stochastic Systems published by Springer in 2002. Basically, the new edition of the book corresponds to its ?rst version. While preparingthiseditionwemadesomeclari?cationsinseveralsectionsandalso corrected the misprints noticed in some formulas. Besides, three new sections have been added to Chapter 2. They are Statistical Properties of Dynamical Chaos, E?ects of Synchronization in Extended Self-Sustained Oscillatory Systems, and Synchronization in Living Systems. The sections indicated re?ect the most interesting results obtained by the authors after publication of the ?rst edition. We hope that the new edition of the book will be of great interest for a widesectionofreaderswhoarealreadyspecialistsorthosewhoarebeginning research in the ?elds of nonlinear oscillation and wave theory, dynamical chaos, synchronization, and stochastic process theory. Saratov, Berlin, and St. Louis V.S. Anishchenko November 2006 A.B. Neiman T.E. Vadiavasova V.V. Astakhov L. Schimansky-Geier Preface to the First Edition Thisbookisdevotedtotheclassicalbackgroundandtocontemporaryresults on nonlinear dynamics of deterministic and stochastic systems. Considerable attentionisgiventothee?ectsofnoiseonvariousregimesofdynamicsystems with noise-induced order. On the one hand, there exists a rich literature of excellent books on n- linear dynamics and chaos; on the other hand, there are many marvelous monographs and textbooks on the statistical physics of far-from-equilibrium andstochasticprocesses.Thisbookisanattempttocombinetheapproachof nonlinear dynamics based on the deterministic evolution equations with the approach of statistical physics based on stochastic or kineticequations. One of our main aims is to show the important role of noise in the organization and properties of dynamic regimes of nonlinear dissipative systems.

From the reviews:

"The emphasis of the book is on applicable theory that will be of use in the interpretation of experimental or numerical results, along with a large number of examples that are drawn from many sources. In order for this book to be useful for a variety of scientists, the authors have tried to keep each part more or less self-contained, have avoided the use of formal mathematical proofs throughout and have included many examples, including many nonlinear circuits." (Mathematical Reviews, 2003h)

"The book is in three parts; the first a tutorial overview, the second on deterministic systems and the third on stochastic systems. The tutorial overview is a whirlwind tour through Lyapunov exponents, homoclinic tangles, chaotic synchronization, stochastic self-sustained oscillations and much more. Such a comprehensive overview is excellent for researchers in Complex Systems who could use it as an extended dictionary. [...] The subsequent two parts of the book provide highly detailed descriptions and numerous references for researchers interested in studying nonlinear dynamical systems in the presence of noise. [...] I highly commend this book to all researchers with a serious interest in Complex Systems." (B.I. Henry, The Physicist, 39/6, 2002)

From the reviews of the second edition:

"This is the second edition of the original version published in 2002 . The topics of this book are many aspects of finite dimensional complex deterministic and stochastic dynamical systems from a physicists' perspective. Its value for mathematicians lies mainly in the fact that it presents an up-to-date account of currently relevant topics in physics. The presentation includes many examples and references mainly to the physics literature." (Fritz Colonius, Zentralblatt MATH, Vol. 1125 (2), 2008)

Texte du rabat

This book is a complete treatise on the theory of nonlinear dynamics of chaotic and stochastic systems. It contains both an exhaustive introduction to the subject as well as a detailed discussion of fundamental problems and research results in a field to which the authors have made important contributions themselves. Despite the unified presentation of the subject, care has been taken to present the material in largely self-contained chapters.
The present book can thus be used either as a textbook by graduate students or as a modern monograph by researchers in this field.

The second edition has been substantially enlarged to include sections on statistical properties of dynamical chaos, on effects of synchronization in oscillatory systems and on synchronization in living systems.

Contenu
Tutorial.- Dynamical Systems.- Fluctuations in Dynamical Systems.- Synchronization of Periodic Systems.- Dynamical Chaos.- Routes to Chaos.- Statistical Properties of Dynamical Chaos.- Synchronization of Chaos.- Effects of Synchronization in Extended Self-Sustained Oscillatory Systems.- Synchronization in Living Systems.- Controlling Chaos.- Reconstruction of Dynamical Systems.- Stochastic Dynamics.- Stochastic Resonance.- Synchronization of Stochastic Systems.- The Beneficial Role of Noise in Excitable Systems.- Noise Induced Transport.