Comptes Rendus
no. 5
Dynamical Systems
Time and entry–exit relation near a planar turning point
[Temps et relation entrée–sortie proche d'un point tournant planaire.]

Présenté par : Jean-Christophe Yoccoz

Peter De Maesschalck1 ; Freddy Dumortier1
1 Limburgs Universitair Centrum, 3590 Diepenbeek, Belgium
Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 359-364.

Suivant l'approche géométrique dans l'étude de problèmes de perturbations singulières dans le plan aux points tournants, et travaillant dans un cadre très général dans lequel apparaissent des solutions canards, nous étudions le temps de passage des orbites proche des points tournants, tout comme la relation entrée–sortie à tel point. Les variétés de solutions canards rencontrées ne sont en général que C0 à un point tournant, ne permettant pas une approche asymptotique classique. L'approche est basée sur l'éclatement et l'utilisation de variétés centrales et de formes normales Ck.

Following the geometric approach for studying singular perturbation problems in the plane at turning points, and considering a very general setting where canard solutions are shown to exist, we study the transition time of orbits passing near the turning point, as well as the entry–exit relation at such turning points. The manifolds of canard solutions are in general only C0 at the turning point, making the classical asymptotic approach impossible. The method involves a (family) blow up of the turning point and the use of Ck-normal forms and center manifolds.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.06.020
Peter De Maesschalck 1 ; Freddy Dumortier 1

1 Limburgs Universitair Centrum, 3590 Diepenbeek, Belgium 
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Peter De Maesschalck; Freddy Dumortier. Time and entry–exit relation near a planar turning point. Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 359-364. doi : 10.1016/j.crma.2004.06.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.06.020/

[1] P. Bonckaert Partially hyperbolic fixed points with constraints, Trans. Amer. Math. Soc., Volume 348 (1996) no. 3, pp. 997-1011

[2] F. Dumortier; P. De Maesschalck Topics on singularities and bifurcations of vector fields, Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II: Math. Phys. Chem., vol. 137, 2004

[3] F. Dumortier, P. De Maesschalck, Time analysis and entry–exit relation near planar turning points, 2004, preprint

[4] F. Dumortier; R. Roussarie Canard cycles and center manifolds, Mem. Amer. Math. Soc., Volume 121 (1996), p. 577

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

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