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]
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 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 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 : https://doi.org/10.5802/aif.2747
Classification : 37G05,  34C20,  37C10
Mots clés : formes normales, partie linéaire nilpotente, normalisation Gevrey
@article{AIF_2012__62_6_2211_0,
     author = {Bonckaert, Patrick and Verstringe, Freek},
     title = {Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part},
     journal = {Annales de l'Institut Fourier},
     pages = {2211--2225},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     doi = {10.5802/aif.2747},
     mrnumber = {3060756},
     zbl = {1278.37044},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2747/}
}
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. http://archive.numdam.org/articles/10.5802/aif.2747/

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