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Comptes Rendus Mathématique
Volume 339, n° 5
pages 359-364 (septembre 2004)
Doi : 10.1016/j.crma.2004.06.020
Received : 17 June 2004 ;  accepted : 22 June 2004
Time and entry-exit relation near a planar turning point
Temps et relation entrée-sortie proche d'un point tournant planaire.
 

Peter De Maesschalck , Freddy Dumortier
Limburgs Universitair Centrum, 3590 Diepenbeek, Belgium 

Abstract

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   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  -normal forms and center manifolds. To cite this article: P. De Maesschalck, F. Dumortier, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.
Résumé

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   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  . Pour citer cet article : P. De Maesschalck, F. Dumortier, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.


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