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The De Gruyter Studies in Mathematical Physics are devoted to the
publication of monographs and high-level texts in mathematical
physics. They cover topics and methods in fields of current
interest, with an emphasis on didactical presentation. The series
will enable readers to understand, apply and develop further, with
sufficient rigor, mathematical methods to given problems in
physics. For this reason, works with a few authors are preferred
over edited volumes. The works in this series are aimed at advanced
students and researchers in mathematical and theoretical physics.
They can also serve as secondary reading for lectures and seminars
at advanced levels.
This book is an introduction to the theory of shadowing of
approximate trajectories in dynamical systems by exact ones. This
is the first book completely devoted to the theory of shadowing. It
shows the importance of shadowing theory for both the qualitative
theory of dynamical systems and the theory of numerical methods.
Shadowing Methods allow us to estimate differences between exact
and approximate solutions on infinite time intervals and to
understand the influence of error terms. The book is intended for
specialists in dynamical systems, for researchers and graduate
students in the theory of numerical methods.
This book is an introduction to main methods and principal results
in the theory of Co(remark: o is upper index!!)-small perturbations
of dynamical systems. It is the first comprehensive treatment of
this topic. In particular, Co(upper index!)-generic properties of
dynamical systems, topological stability, perturbations of
attractors, limit sets of domains are discussed. The book contains
some new results (Lipschitz shadowing of pseudotrajectories in
structurally stable diffeomorphisms for instance). The aim of the
author was to simplify and to "visualize" some basic proofs, so the
main part of the book is accessible to graduate students in pure
and applied mathematics. The book will also be a basic reference
for researchers in various fields of dynamical systems and their
applications, especially for those who study attractors or
pseudotrajectories generated by numerical methods.
Focusing on the theory of shadowing of approximate trajectories
(pseudotrajectories) of dynamical systems, this book surveys recent
progress in establishing relations between shadowing and such basic
notions from the classical theory of structural stability as
hyperbolicity and transversality. Special attention is given to the
study of "quantitative" shadowing properties, such as Lipschitz
shadowing (it is shown that this property is equivalent to
structural stability both for diffeomorphisms and smooth flows),
and to the passage to robust shadowing (which is also equivalent to
structural stability in the case of diffeomorphisms, while the
situation becomes more complicated in the case of flows). Relations
between the shadowing property of diffeomorphisms on their chain
transitive sets and the hyperbolicity of such sets are also
described. The book will allow young researchers in the field of
dynamical systems to gain a better understanding of new ideas in
the global qualitative theory. It will also be of interest to
specialists in dynamical systems and their applications.
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