Bifurcation of relaxation oscillations in dimension two

Abstract

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