© 2013

Normally Hyperbolic Invariant Manifolds

The Noncompact Case

Part of the Atlantis Series in Dynamical Systems book series (ASDS, volume 2)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Jaap Eldering
    Pages 1-33
  3. Jaap Eldering
    Pages 35-74
  4. Jaap Eldering
    Pages 75-140
  5. Jaap Eldering
    Pages 141-150
  6. Back Matter
    Pages 151-189

About this book


This monograph treats normally hyperbolic invariant manifolds, with a focus on noncompactness. These objects generalize hyperbolic fixed points and are ubiquitous in dynamical systems.
First, normally hyperbolic invariant manifolds and their relation to hyperbolic fixed points and center manifolds, as well as, overviews of history and methods of proofs are presented. Furthermore, issues (such as uniformity and bounded geometry) arising due to noncompactness are discussed in great detail with examples.
The main new result shown is a proof of persistence for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. This extends well-known results by Fenichel and Hirsch, Pugh and Shub, and is complementary to noncompactness results in Banach spaces by Bates, Lu and Zeng. Along the way, some new results in bounded geometry are obtained and a framework is developed to analyze ODEs in a differential geometric context.
Finally, the main result is extended to time and parameter dependent systems and overflowing invariant manifolds.


Bounded geometry Dynamical systems Noncompactness Normally hyperbolic invariant manifolds Persistence

Authors and affiliations

  1. 1.Deaprtment of MathematicsUtrecht UniversityUtrechtThe Netherlands

Bibliographic information

  • Book Title Normally Hyperbolic Invariant Manifolds
  • Book Subtitle The Noncompact Case
  • Authors Jaap Eldering
  • Series Title Atlantis Series in Dynamical Systems
  • Series Abbreviated Title Atlantis Ser.Dynamical Systems
  • DOI
  • Copyright Information Atlantis Press and the author 2013
  • Publisher Name Atlantis Press, Paris
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-94-6239-002-7
  • Softcover ISBN 978-94-6239-042-3
  • eBook ISBN 978-94-6239-003-4
  • Edition Number 1
  • Number of Pages XII, 189
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Dynamical Systems and Ergodic Theory
    Mathematics, general
  • Buy this book on publisher's site