Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations
Fractal Dimensions and Infinitely Many Attractors in Dynamics

A self-contained introduction to the classical theory and its generalizations, aimed at mathematicians and scientists working in dynamical systems.

Jacob Palis (Author), Floris Takens (Author)

9780521475723, Cambridge University Press

Paperback, published 5 January 1995

248 pages
22.7 x 15.1 x 1.4 cm, 0.34 kg

"...an interesting monograph on this collection of ideas. It begins with material often covered in graduate texts but quickly moves to the exposition of ideas only available in the original sources. It would be quite suitable for an advanced graduate level course in dynamics and bifurcation theory." John Franks, Bulletin of the American Mathematical Society

This is a self-contained introduction to the classical theory of homoclinic bifurcation theory, as well as its generalizations and more recent extensions to higher dimensions. It is also intended to stimulate new developments, relating the theory of fractal dimensions to bifurcations, and concerning homoclinic bifurcations as generators of chaotic dynamics. To this end the authors finish the book with an account of recent research and point out future prospects. The book begins with a review chapter giving background material on hyperbolic dynamical systems. The next three chapters give a detailed treatment of a number of examples, Smale's description of the dynamical consequences of transverse homoclinic orbits and a discussion of the subordinate bifurcations that accompany homoclinic bifurcations, including Hénon-like families. The core of the work is the investigation of the interplay between homoclinic tangencies and non-trivial basic sets. The fractal dimensions of these basic sets turn out to play an important role in determining which class of dynamics is prevalent near a bifurcation. The authors provide a new, more geometric proof of Newhouse's theorem on the coexistence of infinitely many periodic attractors, one of the deepest theorems in chaotic dynamics. Based on graduate courses, this unique book will be an essential purchase for students and research workers in dynamical systems, and also for scientists and engineers applying ideas from chaos theory and nonlinear dynamics.

Preface
1. Hyperbolicity, stability and sensitive-chaotic dynamical systems
2. Examples of homoclinic orbits in dynamical systems
3. Dynamical consequences of a transverse homoclinic intersection
4. Homoclinic tangencies: cascades of bifurcations, scaling and quadratic maps
5. Cantor sets in dynamics and fractal dimensions
6. Homoclinic bifurcations: fractal dimensions and measure of bifurcation sets
7. Infinitely many sinks and homoclinic tangencies
8. Overview, conjectures and problems - a theory of homoclinic bifurcations - strange attractors
Appendices
References.

Subject Areas: Calculus & mathematical analysis [PBK]