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Normally Hyperbolic Invariant Manifolds

The Noncompact Case

  • Jaap Eldering

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

Introduction

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.

Keywords

Bounded geometry Dynamical systems Noncompactness Normally hyperbolic invariant manifolds Persistence

Authors and affiliations

  • Jaap Eldering
    • 1
  1. 1.Deaprtment of MathematicsUtrecht UniversityUtrechtThe Netherlands

Bibliographic information

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