Comptes Rendus
Ordinary differential equations/Dynamical systems
Canard-cycle transition at a fast–fast passage through a jump point
Comptes Rendus. Mathématique, Volume 352 (2014) no. 1, pp. 27-30.

We consider transitory canard cycles that consist of a generic breaking mechanism, i.e. a Hopf or a jump breaking mechanism, in combination with a fast–fast passage through a jump point. Such cycle separates two types of canard cycles with a different shape. We obtain upper bounds on the number of periodic orbits that can appear near the canard cycle, and this under very general conditions.

On considère des cycles canard transitoires comportant un mécanisme de cassure générique, de type Hopf ou bien de saut, en combinaison avec un passage de type rapide–rapide par un point de saut. De tels cycles séparent deux types de cycles canard de formes différentes. On obtient des bornes supérieures sur le nombre dʼorbites périodiques qui peuvent apparaître près du cycle canard, sous certaines conditions très générales.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.09.002

Peter De Maesschalck 1; Freddy Dumortier 1; Robert Roussarie 2

1 Hasselt University, Martelarenlaan 42, B-3500 Hasselt, Belgium
2 Institut de mathématique de Bourgogne, UMR 5584 du CNRS, université de Bourgogne, BP 47 870, 21078 Dijon cedex, France
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Peter De Maesschalck; Freddy Dumortier; Robert Roussarie. Canard-cycle transition at a fast–fast passage through a jump point. Comptes Rendus. Mathématique, Volume 352 (2014) no. 1, pp. 27-30. doi : 10.1016/j.crma.2013.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.002/

[1] P. De Maesschalck, F. Dumortier, R. Roussarie, Canard cycle transition at a slow–fast passage through a jump point, submitted for publication.

[2] P. De Maesschalck, F. Dumortier, R. Roussarie, Canard cycles from birth to transition, in preparation.

[3] P. De Maesschalck; F. Dumortier; R. Roussarie Cyclicity of common slow–fast cycles, Indag. Math. (N. S.), Volume 22 (2011) no. 3–4, pp. 165-206

[4] F. Dumortier Slow divergence integral and balanced canard solutions, Qual. Theory Dyn. Syst., Volume 10 (2011) no. 1, pp. 65-85

[5] F. Dumortier; R. Roussarie Multiple canard cycles in generalized Liénard equations, J. Differential Equations, Volume 174 (2001) no. 1, pp. 1-29

[6] F. Dumortier; R. Roussarie Birth of canard cycles, Discrete Contin. Dyn. Syst. Ser. S, Volume 2 (2009) no. 4, pp. 723-781

[7] M. Krupa; P. Szmolyan Relaxation oscillation and canard explosion, J. Differential Equations, Volume 174 (2001) no. 2, pp. 312-368

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