Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation

Eric Lombardi; Laurent Stolovitch

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 4, page 659-718
  • ISSN: 0012-9593

Abstract

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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 ℂ 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.

How to cite

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Lombardi, Eric, and 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 43.4 (2010): 659-718. <http://eudml.org/doc/272208>.

@article{Lombardi2010,
abstract = {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 $\mathbb \{C\}^n$, 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.},
author = {Lombardi, Eric, Stolovitch, Laurent},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {differential equations; small divisors; resonances; normal forms},
language = {eng},
number = {4},
pages = {659-718},
publisher = {Société mathématique de France},
title = {Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation},
url = {http://eudml.org/doc/272208},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Lombardi, Eric
AU - Stolovitch, Laurent
TI - Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 4
SP - 659
EP - 718
AB - 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 $\mathbb {C}^n$, 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.
LA - eng
KW - differential equations; small divisors; resonances; normal forms
UR - http://eudml.org/doc/272208
ER -

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