Normally Hyperbolic Invariant Manifolds, 1st Edition

  • Published By:
  • ISBN-10: 9462390037
  • ISBN-13: 9789462390034
  • DDC: 515.352
  • Grade Level Range: College Freshman - College Senior
  • 189 Pages | eBook
  • Original Copyright 2013 | Published/Released June 2014
  • This publication's content originally published in print form: 2013

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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.

Table of Contents

Front Cover.
Other Frontmatter.
Title Page.
Copyright Page.
Series Information.
1: Introduction.
2: Manifolds of Bounded Geometry.
3: Persistence of Noncompact NHIMs.
4: Extension of Results.
Explicit Estimates in the Implicit Function Theorem.
The Nemytskii Operator.
Exponential Growth Estimates.
The Fiber Contraction Theorem.
Nonlinear Variation of Flows.
Riemannian Geometry.