Canard Cycles and Center Manifolds

Download or read book Canard Cycles and Center Manifolds written by Freddy Dumortier and published by American Mathematical Soc.. This book was released on 1996 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.