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Geometric Theory of Discrete Nonautonomous Dynamical Systems

  • Book
  • © 2010

Overview

Authors:
  1. Christian Pötzsche

    1. Centre for Mathematical Sciences, Munich University of Technology, Garching, Germany

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  • Comprehensive approach to discrete dynamical systems
  • Applications to numerical discretizations
  • Extensive invariant manifold theory
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xxiv
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  2. Nonautonomous Dynamical Systems

    • Christian Pötzsche
    Pages 1-36

  3. Nonautonomous Difference Equations

    • Christian Pötzsche
    Pages 37-94

  4. Linear Difference Equations

    • Christian Pötzsche
    Pages 95-185

  5. Invariant Fiber Bundles

    • Christian Pötzsche
    Pages 187-316

  6. Linearization

    • Christian Pötzsche
    Pages 317-343

  7. Back Matter

    Pages 345-405
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Keywords

About this book

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.

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)

Authors and Affiliations

  • Centre for Mathematical Sciences, Munich University of Technology, Garching, Germany

    Christian Pötzsche

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

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2010

  • Softcover ISBN: 978-3-642-14257-4Published: 18 September 2010

  • eBook ISBN: 978-3-642-14258-1Published: 24 August 2010

  • 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

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