Fractal Dimensions for Poincare Recurrences

Informationen zum Autor 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. Zusammenfassung Deals with 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. This book presents rules for action to study mathematical models of real systems. It contains standard theorems of dynamical systems theory.

Autorentext
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.

Klappentext
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.

• Some results 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.
• It is a rare book where rigorous mathematical theory is combined with important physical applications.
• The book presents rules for immediate action to study mathematical models of real systems.

• Together with standard theorems of dynamical systems theory, the book contains some not very well known results that became more and more important for applications.

  Inhalt

  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