Fractal Dimensions for Poincare Recurrences, Volume 2
1st Edition
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Table of Contents
- Introduction
Part 1: Fundamentals - Symbolic Systems
- Geometric Constructions
- Spectrum of Dimensions for Recurrences
Part II: Zero-Dimensional Invariant Sets - Uniformly Hyperbolic Repellers
- Non-Uniformly Hyperbolic Repellers
- The Spectrum for a Sticky Set
- Rhythmical Dynamics
Part III: One-Dimensional Systems - Markov Maps of the Interval
- Suspended Flows
Part IV: Measure Theoretical Results - Invariant Measures
- Dimensional for Measures
- The Variational Principle
Part V: Physical Interpretation and Applications - Intuitive Explanation
- Hamiltonian Systems
- Chaos Synchronization
Part VI: Appendices - Some Known Facts About Recurrences
- Birkhoff's Individual Theorem
- The SMB Theorem
- Amalgamation and Fragmentation
Index
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
Details
- No. of pages:
- 258
- Language:
- English
- Copyright:
- © Elsevier Science 2006
- Published:
- 21st June 2006
- Imprint:
- Elsevier Science
- Hardcover ISBN:
- 9780444521897
- eBook ISBN:
- 9780080462394
Ratings and Reviews
About the Authors
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.
Affiliations and Expertise
Universidad Autonoma de San Luis Potosi, Mexico.
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.
Affiliations and Expertise
Universidad Autonoma de San Luis Potosi, Mexico.
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.
Affiliations and Expertise
Universidad Autonoma de San Luis Potosi, Mexico.
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