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© 2010

Geometric Theory of Discrete Nonautonomous Dynamical Systems

Benefits

  • Comprehensive approach to discrete dynamical systems

  • Applications to numerical discretizations

  • Extensive invariant manifold theory

Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2002)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Christian Pötzsche
    Pages 1-36
  3. Christian Pötzsche
    Pages 37-94
  4. Christian Pötzsche
    Pages 95-185
  5. Christian Pötzsche
    Pages 187-316
  6. Christian Pötzsche
    Pages 317-343
  7. Back Matter
    Pages 345-405

About this book

Introduction

Nonautonomous dynamical systems provide a mathematical framework for temporally changing phenomena, where the law of evolution varies in time due to seasonal, modulation, controlling or even random effects. Our goal is to provide an approach to the corresponding geometric theory of nonautonomous discrete dynamical systems in infinite-dimensional spaces by virtue of 2-parameter semigroups (processes). These dynamical systems are generated by implicit difference equations, which explicitly depend on time. Compactness and dissipativity conditions are provided for such problems in order to have attractors using the natural concept of pullback convergence. Concerning a necessary linear theory, our hyperbolicity concept is based on exponential dichotomies and splittings. This concept is in turn used to construct nonautonomous invariant manifolds, so-called fiber bundles, and deduce linearization theorems. The results are illustrated using temporal and full discretizations of evolutionary differential equations.

Keywords

Exponential dichotomy Invariant fiber bundles Nonautonomous difference equations Nonautonomous dynamical systems Topological linearization difference equation dynamical systems

Authors and affiliations

  1. 1.Centre for Mathematical SciencesMunich University of TechnologyGarchingGermany

Bibliographic information

  • Book Title Geometric Theory of Discrete Nonautonomous Dynamical Systems
  • Authors Christian Pötzsche
  • Series Title Lecture Notes in Mathematics
  • DOI https://doi.org/10.1007/978-3-642-14258-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 2010
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-642-14257-4
  • eBook ISBN 978-3-642-14258-1
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages XXIV, 399
  • Number of Illustrations 15 b/w illustrations, 2 illustrations in colour
  • Topics Dynamical Systems and Ergodic Theory
  • Buy this book on publisher's site
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Reviews

From the reviews:

“The book contains detailed information concerning the two-parameter semigroups defined by a quite general class of difference equations. … The Hartman-Grobman theory also receives considerable attention. … this is a well-written book which will be very useful to the reader interested in the topics which it discusses.” (Russell A. Johnson, Mathematical Reviews, Issue 2012 a)

“The monograph is a rich resource for a consistent theory of nonautonomous difference equations, in particular their stability theory and the connection between linear and nonlinear systems. … The reader … who is interested in a thorough course on the theory of difference equations will benefit from this book which combines summaries on the different topics with precise and new results.” (Jörg Härterich, Zentralblatt MATH, Vol. 1247, 2012)