Bifurcation of relaxation oscillations in dimension two

Pages: 631 - 674, Volume 19, Issue 4, December 2007

doi:10.3934/dcds.2007.19.631       Abstract        Full Text (523.7K)       Related Articles

Freddy Dumortier - Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium (email)
Robert Roussarie - Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S., Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France (email)

Abstract: The paper deals with the bifurcation of relaxation oscillations in two dimensional slow-fast systems. The most generic case is studied by means of geometric singular perturbation theory, using blow up at contact points. It reveals that the bifurcation goes through a continuum of transient canard oscillations, controlled by the slow divergence integral along the critical curve. The theory is applied to polynomial Liénard equations, showing that the cyclicity near a generic coallescence of two relaxation oscillations does not need to be limited to two, but can be arbitrarily high.

Keywords:  Slow-fast system, bifurcation, relaxation oscillation, canard cycle, Liénard equation.
Mathematics Subject Classification:  Primary: 34C05, 34C07, 34C23, 34C26, 34E15.

Received: December 2006;      Revised: June 2007;      Published: September 2007.