Description

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.

Key Features

* 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

Readership

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

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

Details

No. of pages:
258
Language:
English
Copyright:
© 2006
Published:
Imprint:
Elsevier Science
Print ISBN:
9780444521897
Electronic ISBN:
9780080462394

About the editors

Valentin Afraimovich

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.

Edgardo Ugalde

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.

Jesus Urias

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.