Fractal Dimensions for Poincare Recurrences


  • Valentin Afraimovich, Universidad Autonoma de San Luis Potosi, Mexico.
  • Edgardo Ugalde, Universidad Autonoma de San Luis Potosi, Mexico.
  • Jesus Urias, Universidad Autonoma de San Luis Potosi, Mexico.

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.

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Researchers, lecturers and students in Nonlinear, Statistical and Mathematical Physics


Book information

  • Published: June 2006
  • Imprint: ELSEVIER
  • ISBN: 978-0-444-52189-7

Table of Contents

1. Introduction
Part 1: Fundamentals
2. Symbolic Systems3. Geometric Constructions4. Spectrum of Dimensions for Recurrences
Part II: Zero-Dimensional Invariant Sets
5. Uniformly Hyperbolic Repellers6. Non-Uniformly Hyperbolic Repellers7. The Spectrum for a Sticky Set8. Rhythmical Dynamics
Part III: One-Dimensional Systems
9. Markov Maps of the Interval10. Suspended Flows
Part IV: Measure Theoretical Results
11. Invariant Measures12. Dimensional for Measures13. The Variational Principle
Part V: Physical Interpretation and Applications
14. Intuitive Explanation15. Hamiltonian Systems16. Chaos Synchronization
Part VI: Appendices
17. Some Known Facts About Recurrences18. Birkhoff's Individual Theorem19. The SMB Theorem20. Amalgamation and Fragmentation