Normally Hyperbolic Invariant Manifolds in Dynamical Systems

1. Front Matter

   Pages i-ix

   PDF

2. Introduction, Motivation, and Background

   • Stephen Wiggins

   Pages 1-19

3. Background from the Theory of Differentiable Manifolds

   • Stephen Wiggins

   Pages 21-49

4. Persistence of Overflowing Invariant Manifolds— Fenichel’s Theorem

   • Stephen Wiggins

   Pages 51-109

5. The Unstable Manifold of an Overflowing Invariant Manifold

   • Stephen Wiggins

   Pages 111-130

6. Foliations of Unstable Manifolds

   • Stephen Wiggins

   Pages 131-157

7. Miscellaneous Properties and Results

   • Stephen Wiggins

   Pages 159-163

8. Examples

   • Stephen Wiggins

   Pages 165-183

9. Back Matter

   Pages 185-194

   PDF

In the past ten years, there has been much progress in understanding the global dynamics of systems with several degrees-of-freedom. An important tool in these studies has been the theory of normally hyperbolic invariant manifolds and foliations of normally hyperbolic invariant manifolds. In recent years these techniques have been used for the development of global perturbation methods, the study of resonance phenomena in coupled oscillators, geometric singular perturbation theory, and the study of bursting phenomena in biological oscillators. "Invariant manifold theorems" have become standard tools for applied mathematicians, physicists, engineers, and virtually anyone working on nonlinear problems from a geometric viewpoint. In this book, the author gives a self-contained development of these ideas as well as proofs of the main theorems along the lines of the seminal works of Fenichel. In general, the Fenichel theory is very valuable for many applications, but it is not easy for people to get into from existing literature. This book provides an excellent avenue to that. Wiggins also describes a variety of settings where these techniques can be used in applications.

• dynamical systems
• dynamics
• manifold

• Applied Mechanics Department 104-44, California Institute of Technology, Pasadena, USA

  Stephen Wiggins

Policies and ethics