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

Zhihua Ren; Jiazhong Yang

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

Zhihua Ren ¹ ; Jiazhong Yang ¹

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

Comptes Rendus. Mathématique, Volume 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 :

Zhihua Ren ¹ ; Jiazhong Yang ¹

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

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     author = {Zhihua Ren and Jiazhong Yang},
     title = {On the {\protect\emph{C}\protect\textsuperscript{1}} normal forms for hyperbolic vector fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {709--712},
     publisher = {Elsevier},
     volume = {336},
     number = {9},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00173-0},
     language = {en},
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Zhihua Ren; Jiazhong Yang. On the C¹ normal forms for hyperbolic vector fields. Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 709-712. doi : 10.1016/S1631-073X(03)00173-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00173-0/

Bibliographie
Cité par

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[4] I.U. Bronstein; A.Ya. Kopanskii Smooth Invariant Manifolds and Normal Forms, World Scientific, Singapore, 1994

[5] K.T. Chen Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math., Volume 85 (1963), pp. 693-722

[6] V.S. Samovol Linearization of systems of ordinary differential equations in a neighbourhood of invariant toroidal manifolds, Proc. Moscow Math. Soc., Volume 38 (1979), pp. 187-219

[7] S. Sternberg On the structure of local homomorphisms of Euclidean n-space, I, Amer. J. Math., Volume 80 (1958), pp. 623-631

[8] S. Sternberg On the structure of local homomorphisms of Euclidean n-space, II, Amer. J. Math., Volume 81 (1959), pp. 578-605

Cité par Sources :

^☆ The work is supported by NSFC-10271006.

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