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Shadowing in Dynamical Systems

Theory and Applications

  • Book
  • © 2000

Overview

Authors:
  1. Ken Palmer

    1. School of Mathematical & Statistical Sciences, La Trobe University, Bundoora, Australia

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Part of the book series: Mathematics and Its Applications (MAIA, volume 501)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds

    • Ken Palmer
    Pages 1-20

  3. Hyperbolic Sets of Diffeomorphisms

    • Ken Palmer
    Pages 21-55

  4. Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets

    • Ken Palmer
    Pages 57-76

  5. The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms

    • Ken Palmer
    Pages 77-90

  6. Symbolic Dynamics Near a Transversal Homoclinic Point of a Diffeomorphism

    • Ken Palmer
    Pages 91-97

  7. Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and Asymptotic Phase

    • Ken Palmer
    Pages 99-114

  8. Hyperbolic Sets of Ordinary Differential Equations

    • Ken Palmer
    Pages 115-169

  9. Transversal Homoclinic Orbits and Hyperbolic Sets in Differential Equations

    • Ken Palmer
    Pages 171-185

  10. Shadowing Theorems for Hyperbolic Sets of Differential Equations

    • Ken Palmer
    Pages 187-223

  11. Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations

    • Ken Palmer
    Pages 225-239

  12. Numerical Shadowing

    • Ken Palmer
    Pages 241-284

  13. Back Matter

    Pages 285-299
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Keywords

About this book

In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic.
It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic.
Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

Authors and Affiliations

  • School of Mathematical & Statistical Sciences, La Trobe University, Bundoora, Australia

    Ken Palmer

Bibliographic Information

  • Book Title: Shadowing in Dynamical Systems

  • Book Subtitle: Theory and Applications

  • Authors: Ken Palmer

  • Series Title: Mathematics and Its Applications

  • DOI: https://doi.org/10.1007/978-1-4757-3210-8

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media Dordrecht 2000

  • Hardcover ISBN: 978-0-7923-6179-4Published: 29 February 2000

  • Softcover ISBN: 978-1-4419-4827-4Published: 09 December 2010

  • eBook ISBN: 978-1-4757-3210-8Published: 14 March 2013

  • Edition Number: 1

  • Number of Pages: XIV, 300

  • Topics: Ordinary Differential Equations, Numeric Computing, Mathematics, general

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