Fractal Dimensions for Poincare Recurrences

Other | June 1, 2006

byAfraimovich, Valentin, Valentin Afraimovich

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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.



* Portions of the book were published in an article that won the title "month's new hot paper in the field of Mathematics" in May 2004
* Rigorous mathematical theory is combined with important physical applications
* Presents rules for immediate action to study mathematical models of real systems
* Contains standard theorems of dynamical systems theory

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From the Publisher

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 ...

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.

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Format:OtherDimensions:258 pages, 1 × 1 × 1 inPublished:June 1, 2006Publisher:Elsevier ScienceLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0080462391

ISBN - 13:9780080462394

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Extra Content

Table of Contents

1. Introduction
Part 1: Fundamentals
2. Symbolic Systems
3. Geometric Constructions
4. Spectrum of Dimensions for Recurrences
Part II: Zero-Dimensional Invariant Sets
5. Uniformly Hyperbolic Repellers
6. Non-Uniformly Hyperbolic Repellers
7. The Spectrum for a Sticky Set
8. Rhythmical Dynamics
Part III: One-Dimensional Systems
9. Markov Maps of the Interval
10. Suspended Flows
Part IV: Measure Theoretical Results
11. Invariant Measures
12. Dimensional for Measures
13. The Variational Principle
Part V: Physical Interpretation and Applications
14. Intuitive Explanation
15. Hamiltonian Systems
16. Chaos Synchronization
Part VI: Appendices
17. Some Known Facts About Recurrences
18. Birkhoff's Individual Theorem
19. The SMB Theorem
20. Amalgamation and Fragmentation
Index