Canard cycles with two breaking parameters

Freddy Dumortier; Robert Roussarie

doi:10.3934/dcds.2007.17.787

Article Contents

2007, Volume 17, Issue 4: 787-806. Doi: 10.3934/dcds.2007.17.787

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Canard cycles with two breaking parameters

Freddy Dumortier^1, and
Robert Roussarie^2,

1.
Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 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: May 31, 2006

Revised: August 31, 2006

Published: September 2007

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Abstract

We consider two-dimensional slow-fast systems with a layer equation exhibiting canard cycles. The canard cycles under consideration contain both a turning point and a fast orbit connecting two jump points. At both the turning point and the connecting fast orbit we suppose the presence of a parameter permitting generic breaking. Such canard cycles depend on two parameters, that we call phase parameters. We study the relaxation oscillations near the canard cycles by means of a map from the plane of phase parameters to the plane of breaking parameters.

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

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