Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 4, p. 659-718

In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of n , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part S which ensures that if such a perturbation of S is formally conjugate to S then it is also holomorphically conjugate to it. We study the normal form problem relatively to S. We give a condition on S that ensures that there always exists an holomorphic transformation to a normal form. If this condition is not satisfied, we also show, that under some reasonable assumptions, each perturbation of S admits a Gevrey formal normalizing transformation to a Gevrey formal normal form. Finally, we give an exponentially good approximation of the dynamic by a partial normal form.

Dans cet article, nous étudions des germes de champs de vecteurs holomorphes qui sont des perturbations « d’ordres supérieurs » de champs de vecteurs quasi-homogènes au voisinage de l’origine de n , point fixe des champs considérés. Nous définissons une condition « diophantienne » sur le champ quasi-homogène initial S qui assure que si une telle perturbation de S est formellement conjuguée à S alors elle l’est aussi holomorphiquement. Nous étudions le problème de mise sous forme normale relativement à S. Nous donnons une condition suffisante assurant l’existence d’une transformation holomorphe vers une forme normale. Lorsque cette condition n’est pas satisfaite, nous montrons néanmoins, sous une condition raisonnable, l’existence d’une normalisation formelle Gevrey vers une forme normale Gevrey. Enfin, nous montrons l’existence d’une approximation exponentiellement bonne de la dynamique par une forme normale partielle.

DOI : https://doi.org/10.24033/asens.2131
Classification:  37F50,  37F75,  37J40,  37J15,  34F15,  34C20,  34M35
Keywords: differential equations, small divisors, resonances, normal forms
@article{ASENS_2010_4_43_4_659_0,
     author = {Lombardi, Eric and Stolovitch, Laurent},
     title = {Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {4},
     year = {2010},
     pages = {659-718},
     doi = {10.24033/asens.2131},
     zbl = {1202.37071},
     mrnumber = {2722512},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2010_4_43_4_659_0}
}
Lombardi, Eric; Stolovitch, Laurent. Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 4, pp. 659-718. doi : 10.24033/asens.2131. http://www.numdam.org/item/ASENS_2010_4_43_4_659_0/

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