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, Zhihua1 ; Yang, Jiazhong1
1 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China
Comptes Rendus. Mathématique, Tome 336 (2003) no. 9, pp. 709-712.

Etant donnés deux germes de champs de vecteurs hyperboliques définis par des équations différentielles autonomes x ˙= Ax +⋯ et y ˙= By +⋯, où x,y∈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 x ˙= Ax +⋯ and y ˙= By +⋯, where x,y∈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 :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00173-0
Ren, Zhihua 1 ; Yang, Jiazhong 1

1 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China 
@article{CRMATH_2003__336_9_709_0,
     author = {Ren, Zhihua and Yang, Jiazhong},
     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},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00173-0/}
}
TY  - JOUR
AU  - Ren, Zhihua
AU  - Yang, Jiazhong
TI  - On the C1 normal forms for hyperbolic vector fields
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 709
EP  - 712
VL  - 336
IS  - 9
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00173-0/
DO  - 10.1016/S1631-073X(03)00173-0
LA  - en
ID  - CRMATH_2003__336_9_709_0
ER  - 
%0 Journal Article
%A Ren, Zhihua
%A Yang, Jiazhong
%T On the C1 normal forms for hyperbolic vector fields
%J Comptes Rendus. Mathématique
%D 2003
%P 709-712
%V 336
%N 9
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00173-0/
%R 10.1016/S1631-073X(03)00173-0
%G en
%F CRMATH_2003__336_9_709_0
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/

[1] Belitskii, G. Normal Forms, Invariant, and Local Mappings, Naukova Dumka, Kiev, 1979

[2] P. Bonckaert, V. Naudot, J. Yang, Time logarithmic transformations near a resonant hyperbolic equilibrium point, C. R. Acad. Sci. Paris, to appear

[3] Bruno, A. Local Methods in Nonlinear Differential Equations, Springer-Verlag, 1989

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

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

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

[8] Sternberg, S. 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.