Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics (Cambridge Studies in Advanced Mathematics, Series Number 35)

A self-contained introduction to the classical theory and its generalizations, aimed at mathematicians and scientists working in dynamical systems. Contents Preface 0 - Hyperbolicity, stability and sensitive chaotic dynamical systems §1 Hyperbolicity and stability §2 Sensitive chaotic dynamics 1 - Examples of homoclinic orbits in dynamical systems §1 Homoclinic orbits in a deformed linear map §2 The pendulum §3 The horseshoe §4 A homoclinic bifurcation §5 Concluding remark 2 - Dynamical consequences of a transverse homoclinic intersection §1 Description of the situation - linearizing coordinates and a special domain R §2 The maximal invariant subset of R - topological analysis §3 The maximal invariant subset of R - hyperbolicity and invariant foliations §4 The maximal invariant subset of R - structure §5 Conclusions for the dynamics near a transverse homoclinic orbit §6 Homoclinic points of periodic orbits §7 Transverse homoclinic intersections in arbitrary dimensions §8 Historical note 3 - Homoclinic tangencies: cascades of bifurcations, scaling and quadratic maps §1 Cascades of homoclinic tangencies §2 Saddle-node and period doubling bifurcations §3 Cascades of period doubling bifurcations and sinks §4 Homoclinic tangencies, scaling and quadratic maps 4 - Cantor sets in dynamics and fractal dimensions §1 Dynamically defined Cantor sets §2 Numerical invariants of Cantor sets §3 Local invariants and continuity 5 - Homoclinic bifurcations: fractal dimensions and measure of bifurcation sets § 1 Construction of bifurcating families of diffeomorphisms §2 Homoclinic tangencies with bifurcation set of small relative measure - statement of the results §3 Homoclinic tangencies with bifurcation set of small relative measure - idea of proof §4 Heteroclinic cycles and further results on measure of bifurcation sets 6 - Infinitely many sinks and homoclinic tangencies § 1 Persistent tangencies §2 The tent map and the logistic map §3 Henon-like diffeomorphisms §4 Separatrices of saddle points for diffeomorphisms near a homoclinic tangency §5 Proof of the main result §6 Sensitive chaotic orbits near a homoclinic tangency 7 - Overview, conjectures and problems - a theory of homoclinic bifurcations - strange attractors §1 Homoclinic bifurcations and nonhyperbolic dynamics §2 Strange attractors §3 Summary, further results and problems Appendix 1 - Hyperbolicity: stable manifolds and foliations Appendix 2 - Markov partitions Appendix 3 - On the shape of some strange attractors Appendix 4 - Infinitely many sinks in one-parameter families of diffeomorphisms Appendix 5 - Hyperbolicity and the creation of homoclinic orbits, reprinted from Annals of Mathematics 125 (1987) References Index

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. 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 co-existence of infinitely many periodic attractors, one of the deepest theorems in chaotic dynamics.

In this chapter we give background information and references to the literature concerning basic notions in dynamical systems that play an important role in our study of homoclinic bifurcations. 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.