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Random Dynamical Systems

  • Ludwig Arnold

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages I-XV
  2. Random Dynamical Systems and Their Generators

    1. Front Matter
      Pages 1-1
    2. Ludwig Arnold
      Pages 3-47
    3. Ludwig Arnold
      Pages 49-107
  3. Multiplicative Ergodic Theory

    1. Front Matter
      Pages 109-109
    2. Ludwig Arnold
      Pages 201-233
  4. Smooth Random Dynamical Systems

    1. Front Matter
      Pages 303-303
    2. Ludwig Arnold
      Pages 305-403
    3. Ludwig Arnold
      Pages 405-463
    4. Ludwig Arnold
      Pages 465-531
  5. Back Matter
    Pages 533-588

About this book

Introduction

This book is the first systematic presentation of the theory of random dynamical systems, i.e. of dynamical systems under the influence of some kind of randomness. The theory comprises products of random mappings as well as random and stochastic differential equations. The author's approach is based on Oseledets'multiplicative ergodic theorem for linear random systems, for which a detailed proof is presented. This theorem provides us with a random substitute of linear algebra and hence can serve as the basis of a local theory of nonlinear random systems. In particular, global and local random invariant manifolds are constructed and their regularity is proved. Techniques for simplifying a system by random continuous or smooth coordinate tranformations are developed (random Hartman-Grobman theorem, random normal forms). Qualitative changes in families of random systems (random bifurcation theory) are also studied. A dynamical approach is proposed which is based on sign changes of Lyapunov exponents and which extends the traditional phenomenological approach based on the Fokker-Planck equation. Numerous instructive examples are treated analytically or numerically. The main intention is, however, to present a reliable and rather complete source of reference which lays the foundations for future works and applications.

Keywords

Kozykel Markov Measure Transformation cocycles glatte Ergodentheorie linear algebra multiplicative ergodic theory multiplikative Ergodentheorie random dynamical systems smooth ergodic theory stochastic bifurcation theory stochastische Bifurkationstheorie zufällige dynamische Systeme

Authors and affiliations

  • Ludwig Arnold
    • 1
  1. 1.Institute for Dynamical SystemsUniversity of BremenBremenGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-12878-7
  • Copyright Information Springer-Verlag Berlin Heidelberg 1998
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08355-6
  • Online ISBN 978-3-662-12878-7
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site
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