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October  2007, 17(4): 787-806. doi: 10.3934/dcds.2007.17.787

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, France

Received  June 2006 Revised  September 2006 Published  January 2007

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.
Keywords: bifurcation, canard cycle, Liénard equation., Slow-fast system.
Mathematics Subject Classification: Primary: 34C05, 34C07; Secondary: 34C23, 34C26, 34E1.
Citation: Freddy Dumortier, Robert Roussarie. Canard cycles with two breaking parameters. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 787-806. doi: 10.3934/dcds.2007.17.787
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