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Nonlinear dynamics. Integrability, chaos and patterns. (English) Zbl 1038.37001
Berlin: Springer (ISBN 3-540-43908-0/hbk). xx, 619 p. (2003).
The book is an extensive treatise of nonlinear dynamical systems with emphasize on the concepts of chaos, integrability and patterns. The first part (about half of the text taking nearly 300 pages in 9 chapters) is devoted to finite-dimensional dynamics and chaos. Starting from the basic qualitative results (linear and nonlinear oscillations, Poincaré-Bendixson theory, attractors and stability), continuing with elementary bifurcations, this part culminates in a thorough discussion of chaos in both dissipative and conservative systems. Discrete as well as continuous-time systems are considered. Various routes to chaos (period doubling, quasiperiodic route, intermittency), criteria for chaos (in terms of Lyapunov exponents, power spectrum and autocorrelation function) and numerous examples of chaos are presented. The last chapter in this part includes more advanced topics, such as time series analysis, stochastic resonance, control and synchronization of chaos and quantum chaos.
The next 5 chapters, 10–14, deal with integrable systems. The first one is concerned with finite-dimensional integrable systems, and the remaining ones with solitonic systems. A major portion of the text is devoted to the Korteweg-de Vries equation; other equations, like nonlinear Schrödinger and sine-Gordon equations are discussed briefly. The methods of inverse scattering transform, Hirota’s bilinearization and Bäcklund transforms are presented.
Chapter 15 contains a discussion of reaction-diffusion systems and spatio-temporal patterns. Potential applications of chaos and solitons are included in Chapter 16. Ten appendices on giving more details of some techniques are included.
It was not the authors’ intention to present a detailed rigorous treatment of nonlinear dynamics; indeed, proofs are generally not included. On the other hand, the book contains numerous examples and exercises divided in two groups by their difficulty. The selection of the material also indicates that the book is oriented more toward researchers in applied sciences than in mathematical theory.

MSC:
37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
35K57 Reaction-diffusion equations
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
81Q50 Quantum chaos
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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