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Mathematics - Dynamical Systems & Differential Equations | Normally Hyperbolic Invariant Manifolds - The Noncompact Case

Normally Hyperbolic Invariant Manifolds

The Noncompact Case

Eldering, Jaap

2013, XII, 189 p. 28 illus.

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  • A gentle introduction: examples, history, overview of methods
  • Bridges nonlinear dynamics and differential geometry
  •  Includes various new results in bounded geometry 
  • Completely worked out persistence proof using the Perron method
  • Multiple appendices with background material

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.

Content Level » Research

Keywords » Bounded geometry - Dynamical systems - Noncompactness - Normally hyperbolic invariant manifolds - Persistence

Related subjects » Dynamical Systems & Differential Equations - Mathematics

Table of contents 

Introduction.- Manifolds of bounded geometry.- Persistence of noncompact NHIMs.- Extension of results.

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