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Over the last four decades there has been extensive development in
the theory of dynamical systems. This book aims at a wide audience
where the first four chapters have been used for an undergraduate
course in Dynamical Systems. Material from the last two chapters
and from the appendices has been used quite a lot for master and
PhD courses. All chapters are concluded by an exercise section. The
book is also directed towards researchers, where one of the
challenges is to help applied researchers acquire background for a
better understanding of the data that computer simulation or
experiment may provide them with the development of the theory.
Over the last four decades there has been extensive development in
the theory of dynamical systems. This book aims at a wide audience
where the first four chapters have been used for an undergraduate
course in Dynamical Systems. Material from the last two chapters
and from the appendices has been used quite a lot for master and
PhD courses. All chapters are concluded by an exercise section. The
book is also directed towards researchers, where one of the
challenges is to help applied researchers acquire background for a
better understanding of the data that computer simulation or
experiment may provide them with the development of the theory.
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 Henon-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.
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