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.
- Book Title Normally Hyperbolic Invariant Manifolds
- Book Subtitle The Noncompact Case
- Series Title Atlantis Series in Dynamical Systems
- Series Abbreviated Title Atlantis Ser.Dynamical Systems
- DOI https://doi.org/10.2991/978-94-6239-003-4
- 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
Dynamical Systems and Ergodic Theory
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