Gevrey normal form for unfoldings of nilpotent contact points of planar slow-fast systems

De Maesschalck, Peter; Doan, Thai Son

Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2597-2621.

On présente des formes normales pour les déploiements de points de contact nilpotents des systèmes lent-rapide dans le plan. Les formes normales sont utiles pour traiter des points de contact reguliers et singuliers (y compris les points tournant). Pour les points de contact reguliers, on obtient une forme normale de type Liénard, tandis que pour les points de contact singuliers, la forme normale est de type Liénard sauf une erreur exponentiellement petite. Nos techniques sont basées sur des estimations de Gevrey des séries formelles et la sommation de Gevrey. Il s’agit d’une extension des résultats connus, basée sur une version de Gevrey de la théorème de préparation de Levinson.

We present normal forms for unfoldings of nilpotent contact points of slow-fast systems in the plane. The normal forms are useful in the treatment of regular and singular contact points (including turning points). For regular contact points, we obtain a normal form of Liénard type, while for singular contact points, the normal form is of Liénard type up to exponentially small error. Our techniques are based on Gevrey estimates on formal power series and Gevrey summation. This extension of earlier results is based on a Gevrey version of Levinson’s preparation theorem.

Classification : 34D14, 34A26, 34M30, 34M60
Keywords: singular perturbations, slow-fast vector field, normal forms, Gevrey asymptotics, Liénard system
Mot clés : perturbation singulière, champs de vecteurs lent-rapide, formes normales, asymptotique Gevrey, systèmes Liénard

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