Canard cycles and center manifolds
About this Title
Freddy Dumortier and Robert Roussarie
Publication: Memoirs of the American Mathematical Society
Publication Year 1996: Volume 121, Number 577
ISBNs: 978-0-8218-0443-8 (print); 978-1-4704-0162-7 (online)
MathSciNet review: 1327208
MSC (1991): Primary 34E10; Secondary 34C23, 58F21
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.
Graduate students, mathematicians, physicists, and engineers interested in ordinary differential equations, specifically singular perturbation problems.
Table of Contents
- 1. Statement of the result: The “canard phenomenon” for the singular Van der Pol equation
- 2. Global desingularization
- 3. Foliations by center manifolds
- 4. The canard phenomenon