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Canard Cycles and Center Manifolds
Freddy Dumortier, Limburgs Universitair Centrum, Diepenbeek, Belgium, and Robert Roussarie, Université de Bourgogne, Dijon, France
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Memoirs of the American Mathematical Society
1996; 100 pp; softcover
Volume: 121
ISBN-10: 0-8218-0443-X
ISBN-13: 978-0-8218-0443-8
List Price: US$41
Individual Members: US$24.60
Institutional Members: US$32.80
Order Code: MEMO/121/577
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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.

Readership

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

Table of Contents

  • Statement of the result: the "canard phenomenon" for the singular Van der Pol equation
  • Global desingularization
  • Foliations by center manifolds
  • The canard phenomenon
  • References
  • Appendix: on the proof of theorem 18 (by Chengzhi Li)
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