no. 6
Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part
[Formes normales avec reste exponentiellement petit et normalisation Gevrey pour les champs de vecteurs avec une partie linéaire nilpotente]
Bonckaert, Patrick1 ; Verstringe, Freek1
1 Universiteit Hasselt Agoralaan Gebouw D 3590 Diepenbeek (Belgium)
Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2211-2225.

Nous investiguons la convergence/divergence de la forme normale d’une singularité d’un champ de vecteurs de ℂ n avec une partie linéaire nilpotente. Nous prouvons que chaque champ de vecteurs Gevrey-α avec une partie linéaire nilpotente peut être réduit à une forme normale Gevrey-1+α en utilisant une transformation Gevrey-1+α. Nous prouvons également que si on arrête la procédure de normalisation à un certain ordre optimal, le reste de la forme normale devient exponentiellement petit.

We explore the convergence/divergence of the normal form for a singularity of a vector field on ℂ n with nilpotent linear part. We show that a Gevrey-α vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey-1+α type with the use of a Gevrey-1+α transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

DOI : 10.5802/aif.2747
Classification : 37G05, 34C20, 37C10
Keywords: normal forms, nilpotent linear part, representation theory, Gevrey normalization
Mot clés : formes normales, partie linéaire nilpotente, normalisation Gevrey

Bonckaert, Patrick 1 ; Verstringe, Freek 1

1 Universiteit Hasselt Agoralaan Gebouw D 3590 Diepenbeek (Belgium) 
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     title = {Normal forms with exponentially small remainder and {Gevrey} normalization for vector fields with a nilpotent linear part},
     journal = {Annales de l'Institut Fourier},
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Bonckaert, Patrick; Verstringe, Freek. Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2211-2225. doi : 10.5802/aif.2747. https://aif.centre-mersenne.org/articles/10.5802/aif.2747/

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