Fractal Dimensions Poincare Recurr
Valentin Afraimovich, Edgardo Ugalde, Jesus Urias, J. Urias, E. UgaldeBooks.org participates in affiliate programs including Bookshop.org and the Amazon Services LLC Associates Program. We may earn a commission from qualifying purchases made through links on this page, at no additional cost to you.
Overview
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
Synopsis
In this book, the authors introduce and study a new characteristic of complexity of motions in dynamical systems that has recently appeared. The characteristic measures an average return time and can be expressed in the framework of the dimension theory in dynamical systems.To show it, the authors summarized the current state of the modern dynamical systems theory and the theory of fractal dimensions. They use ideas and notions from symbolic and smooth dynamics, like topological entropy and topological pressure, Lyapunov exponents, etc. Therefore the book can serve as a text for special topic courses for graduate students.
Being written on a high level of mathematical rigorousness, the book can be useful for people from applied sciences as well. An “euristic” chapter and examples of applications of the developed theory show how to use dimensions for Poincare recurrences in specific problems of nonlinear dynamics. In particular, an impressive result is related to distributions of Poincare recurrences in a non purely chaotic Hamiltonian systems when the asymptotic law has polynomial tails. The exponent in the law is expressed by means of the dimension for Poincare recurremces.
This dimension can also serve very effectivelly to characterize synchronization regimes in coupled chaotic systems. These two applications provide a complete justification of the introduced chractaeristics.
The authors have written this book in a clear and detailed way, the exposition is proper and scrupulous.