Overview
- 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
Part of the book series: Atlantis Studies in Dynamical Systems (ASDS, volume 2)
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Table of contents (4 chapters)
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Front Matter
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Back Matter
Keywords
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.
Authors and Affiliations
Bibliographic Information
Book Title: Normally Hyperbolic Invariant Manifolds
Book Subtitle: The Noncompact Case
Authors: Jaap Eldering
Series Title: Atlantis Studies in Dynamical Systems
DOI: https://doi.org/10.2991/978-94-6239-003-4
Publisher: Atlantis Press Paris
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Atlantis Press and the author 2013
Hardcover ISBN: 978-94-6239-002-7Published: 26 August 2013
Softcover ISBN: 978-94-6239-042-3Published: 03 October 2015
eBook ISBN: 978-94-6239-003-4Published: 17 August 2013
Series ISSN: 2213-3526
Series E-ISSN: 2213-3534
Edition Number: 1
Number of Pages: XII, 189
Topics: Dynamical Systems and Ergodic Theory, Mathematics, general
