Canard cycle transition at a slow–fast passage through a jump point

Peter De Maesschalck; Freddy Dumortier; Robert Roussarie

doi:10.1016/j.crma.2014.02.008

Partial differential equations

Canard cycle transition at a slow–fast passage through a jump point

Presented by: the Editorial Board

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

Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 317-320.

Abstract
Résumé

We introduce transitory canard cycles for slow–fast vector fields in the plane. Such cycles separate “canards without head” and “canards with head”, like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle under whatever condition on the related slow divergence integral I, including the challenging case $I = 0$ .

On introduit des cycles canard transitoires pour les champs de vecteurs lents–rapides du plan. De tels cycles font la transition entre des « canards sans tête » et des « canards avec tête », comme par exemple dans l'équation de Van der Pol. On obtient des bornes supérieures optimales pour le nombre des orbites périodiques qui peuvent apparaître près du cycle canard transitoire, quelles que soient les conditions sur l'intégrale de divergence lente I associée, ce qui inclut le cas difficile $I = 0$ .

Received: 2013-04-03
Accepted: 2014-02-17
Published online: 2014-03-13

DOI: 10.1016/j.crma.2014.02.008

Author's affiliations:

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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     author = {Peter De Maesschalck and Freddy Dumortier and Robert Roussarie},
     title = {Canard cycle transition at a slow{\textendash}fast passage through a jump point},
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Peter De Maesschalck; Freddy Dumortier; Robert Roussarie. Canard cycle transition at a slow–fast passage through a jump point. Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 317-320. doi : 10.1016/j.crma.2014.02.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.02.008/

References
Cited by

[1] E. Benoit Équations différentielles: relation entrée–sortie, C. R. Acad. Sci. Paris, Ser. I, Volume 293 (1981) no. 5, pp. 293-296

[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] P. De Maesschalck; F. Dumortier; R. Roussarie Canard-cycle transition at a fast–fast passage through a jump point, C. R. Acad. Sci. Paris, Ser. I, Volume 352 (2014) no. 1, pp. 27-30

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

[6] F. Dumortier; R. Roussarie Canard cycles and center manifolds, Mem. Amer. Math. Soc., Volume 121 (1996) no. 577 (x+100. With an appendix by Li Chengzhi)

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

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

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