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This book is for those interested in dynamical systems. It assumes
a solid undergraduate training in mathematics. Geometrical methods
are developed to study the process of iteration, which involves
taking the output of a function and feeding it back as input.
Iteration processes are used to produce fractals and wavelets, and
to numerically approximate solutions to ordinary and partical
differential equations. Each iteration procedure generates a
discrete dynamical system. These systems are at the heart of many
numerical algorithms. Essentially all mathematical models of
evolving physical systems can be viewed as discrete dynamical
systems. This book attempts to present the fundamental ideas of
discrete dynamical systems as clearly and geometrically as
possible. Illustrative examples of dynamical systems are presented
in the first chapter. The second chapter gives a review of the
typology of metric spaces. The third presents basic results and
establishes a philosophy of dynamics which is strongly influenced
by the work of Charles Conley. The stable manifold and local
structural stability theorems are presented in the fourth chapter.
Invariant sets and isolating blocks are defined in the fifth. The
sixth develops what is called the Conley Index in the context of
discrete dynamics, and the final chpater covers measure-preserving
and symplectic maps. The book would be suitable for use as a main
text for a graduate course in dynamical systems, and as a reference
for engineers and scientists.
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