On the <i>C</i><sup>1</sup> normal forms for hyperbolic vector fields

Ren, Zhihua; Yang, Jiazhong

doi:10.1016/S1631-073X(03)00173-0

Ordinary Differential Equations

On the C¹ normal forms for hyperbolic vector fields
[Sur la classification C¹ des champs de vecteurs hyperboliques]

Présenté par : Bernard Malgrange

Ren, Zhihua ¹ ; Yang, Jiazhong ¹

¹ LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China

Comptes Rendus. Mathématique, Tome 336 (2003) no. 9, pp. 709-712.

Résumé
Abstract

Etant donnés deux germes de champs de vecteurs hyperboliques définis par des équations différentielles autonomes $\dot{x} = Ax + \dots$ et $\dot{y} = By + \dots$ , où $x, y \in R^{n}$ , A et B sont des matrices d'ordre n, on démontre que, sous certaines conditions algébriques sur les valeurs propres des matrices et des conditions de non dégénérescensce des terms nonlinéaires, ils sont au moins C¹ conjuqués si et seulement si A et B sont semblables.

Given two germs of hyperbolic vector fields associated to autonomous ordinary differential equations $\dot{x} = Ax + \dots$ and $\dot{y} = By + \dots$ , where $x, y \in R^{n}$ , and A and B are n×n matrices, we prove that under some algebraic conditions on the eigenvalues of the matrices and genericity condition on the nonlinear terms, they are at least C¹ conjugate if and only if A and B are similar.

Reçu le : 2003-03-19
Accepté le : 2003-03-19
Publié le : 2003-04-30

DOI : 10.1016/S1631-073X(03)00173-0

Affiliations des auteurs :

Ren, Zhihua ¹ ; Yang, Jiazhong ¹

¹ LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China

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     title = {On the {\protect\emph{C}\protect\textsuperscript{1}} normal forms for hyperbolic vector fields},
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Ren, Zhihua; Yang, Jiazhong. On the C¹ normal forms for hyperbolic vector fields. Comptes Rendus. Mathématique, Tome 336 (2003) no. 9, pp. 709-712. doi : 10.1016/S1631-073X(03)00173-0. http://www.numdam.org/articles/10.1016/S1631-073X(03)00173-0/

Bibliographie
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Cité par Sources :

^☆ The work is supported by NSFC-10271006.