Bifurcation of relaxation oscillations in dimension two

Freddy Dumortier; Robert Roussarie

doi:10.3934/dcds.2007.19.631

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2007, Volume 19, Issue 4: 631-674. Doi: 10.3934/dcds.2007.19.631

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Bifurcation of relaxation oscillations in dimension two

Freddy Dumortier¹ and
Robert Roussarie^2,

1.
Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek
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: October 2007

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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.

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Mathematics Subject Classification: Primary: 34C05, 34C07, 34C23, 34C26, 34E15.

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