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.