© 2000

Shadowing in Dynamical Systems

Theory and Applications


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

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.


Manifold Morphism calculus computer dynamical systems dynamische Systeme equation ordinary differential equation proof theorem

Authors and affiliations

  1. 1.School of Mathematical & Statistical SciencesLa Trobe UniversityBundooraAustralia

Bibliographic information

  • Book Title Shadowing in Dynamical Systems
  • Book Subtitle Theory and Applications
  • Authors K.J. Palmer
  • Series Title Mathematics and Its Applications
  • DOI
  • Copyright Information Springer-Verlag US 2000
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-7923-6179-4
  • Softcover ISBN 978-1-4419-4827-4
  • eBook ISBN 978-1-4757-3210-8
  • Edition Number 1
  • Number of Pages XIV, 300
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Ordinary Differential Equations
    Numeric Computing
    Mathematics, general
  • Buy this book on publisher's site