**Memoirs of the American Mathematical Society**

1996;
100 pp;
Softcover

MSC: Primary 34; 58;

Print ISBN: 978-0-8218-0443-8

Product Code: MEMO/121/577

List Price: $44.00

Individual Member Price: $26.40

**Electronic ISBN: 978-1-4704-0162-7
Product Code: MEMO/121/577.E**

List Price: $44.00

Individual Member Price: $26.40

# Canard Cycles and Center Manifolds

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*Freddy Dumortier; Robert Roussarie*

In this book, the “canard phenomenon” occurring in Van der Pol's equation \(\epsilon \ddot x+(x^2+x)\dot x+x-a=0\) is studied. For sufficiently small \(\epsilon >0\) and for decreasing \(a\), the limit cycle created in a Hopf bifurcation at \(a = 0\) stays of “small size” for a while before it very rapidly changes to “big size”, representing the typical relaxation oscillation. The authors give a geometric explanation and proof of this phenomenon using foliations by center manifolds and blow-up of unfoldings as essential techniques. The method is general enough to be useful in the study of other singular perturbation problems.

#### Table of Contents

# Table of Contents

## Canard Cycles and Center Manifolds

- Table of Contents vii8 free
- 1 Statement of the result: the "canard phenomenon" for the singular Van der Pol equation 314 free
- 2 Global desingularization 1021
- 3 Foliations by center manifolds 3647
- 4 The canard phenomenon 7384
- References 95106
- Appendix: on the proof of theorem 18 97108

#### Readership

Graduate students, mathematicians, physicists, and engineers interested in ordinary differential equations, specifically singular perturbation problems.