Fractal Dimensions for Poincare Recurrences

The authors started to work on the subject in 1997 because of requirements in nonlinear dynamics to find out quantities that could measure different behavior in time in dynamical systems. They introduced and studied fractal dimensions for Poincare recurrences that appeared to be new, useful characteristics of complexity of dynamics.The authors started to work on the subject in 1997 because of requirements in nonlinear dynamics to find out quantities that could measure different behavior in time in dynamical systems. They introduced and studied fractal dimensions for Poincare recurrences that appeared to be new, useful characteristics of complexity of dynamics.The authors started to work on the subject in 1997 because of requirements in nonlinear dynamics to find out quantities that could measure different behavior in time in dynamical systems. They introduced and studied fractal dimensions for Poincare recurrences that appeared to be new, useful characteristics of complexity of dynamics.

This book is devoted to an important branch of the dynamical systems theory : the study of the fine (fractal) structure of Poincare recurrences -instants of time when the system almost repeats its initial state. The authors were able to write an entirely self-contained text including many insights and examples, as well as providing complete details of proofs. The only prerequisites are a basic knowledge of analysis and topology. Thus this book can serve as a graduate text or self-study guide for courses in applied mathematics or nonlinear dynamics (in the natural sciences). Moreover, the book can be used by specialists in applied nonlinear dynamics following the way in the book. The authors applied the mathematical theory developed in the book to two important problems: distribution of Poincare recurrences for nonpurely chaotic Hamiltonian systems and indication of synchronization regimes in coupled chaotic individual systems.

1. IntroductionPart 1: Fundamentals2. Symbolic Systems3. Geometric Constructions4. Spectrum of Dimensions for RecurrencesPart II: Zero-Dimensional Invariant Sets5. Uniformly Hyperbolic Repellers6. Non-Uniformly Hyperbolic Repellers7. The Spectrum for a Sticky Set8. Rhythmical DynamicsPart III: One-Dimensional Systems9. Markov Maps of the Interval10. Suspended FlowsPart IV: Measure Theoretical Results11. Invariant Measures12. Dimensional for Measures13. The Variational PrinciplePart V: Physical Interpretation and Applications14. Intuitive Explanation15. Hamiltonian Systems16. Chaos SynchronizationPart VI: Appendices17. Some Known Facts About Recurrences18. Birkhoff's Individual Theorem19. The SMB Theorem20. Amalgamation and FragmentationIndex