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December  2007, 19(4): 631-674. doi: 10.3934/dcds.2007.19.631

Bifurcation of relaxation oscillations in dimension two

Freddy Dumortier 1, and  Robert Roussarie 2,

1. 

Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium
2. 

Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S., Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex

Received  December 2006 Revised  June 2007 Published  September 2007

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: canard cycle, Slow-fast system, Liénard equation., relaxation oscillation, bifurcation.
Mathematics Subject Classification: Primary: 34C05, 34C07, 34C23, 34C26, 34E1.
Citation: Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631
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