Extension of invariant manifolds and applications

Brahim Abbaci

doi:10.1016/j.crma.2005.10.020

Dynamical Systems/Ordinary Differential Equations

Extension of invariant manifolds and applications

Presented by: Étienne Ghys

Brahim Abbaci ¹

¹Université des sciences et de la technologie Houari-Boumèdiene, faculté de mathématiques, BP 32, El Alia, 16111 Alger, Algeria

Comptes Rendus. Mathématique, Volume 341 (2005) no. 12, pp. 755-759

Abstract (Original language)
Résumé

We state an extension theorem for invariant manifolds of diffeomorphisms near a ‘normally hyperbolic’ invariant torus. We apply this result in particular to the resolution of equations $L_{U_{j}} (f) = θ_{j}$ ( $1 ⩽ j ⩽ n - 1$ ) where the $U_{j}$ 's are linear diagonal vector fields and the $θ_{j}$ 's are germs at 0 of smooth functions on $R^{n}$ .

Nous énonçons un théorème de prolongement de variété invariante et nous en donnons une application à la résolution des équations $L_{U_{j}} (f) = θ_{j}$ ( $1 ⩽ j ⩽ n - 1$ ) où les $U_{j}$ sont des champs de vecteurs linéaires diagonaux et les $θ_{j}$ des germes en 0 de fonctions $C^{\infty}$ de $R^{n}$ vérifiant certaines conditions.

Received: 2005-10-01
Accepted: 2005-10-19
Published online: 2005-11-18

DOI: 10.1016/j.crma.2005.10.020

Author's affiliations:

Brahim Abbaci ¹

¹ Université des sciences et de la technologie Houari-Boumèdiene, faculté de mathématiques, BP 32, El Alia, 16111 Alger, Algeria

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     author = {Brahim Abbaci},
     title = {Extension of invariant manifolds and applications},
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     number = {12},
     doi = {10.1016/j.crma.2005.10.020},
     language = {en},
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Brahim Abbaci. Extension of invariant manifolds and applications. Comptes Rendus. Mathématique, Volume 341 (2005) no. 12, pp. 755-759. doi: 10.1016/j.crma.2005.10.020

References
Cited by

[1] B. Abbaci, Variétés invariantes et applications, Thèse, Université Paris 7, 2001

[2] B. Abbaci, An extension theorem for invariant manifold and some applications, in preparation

[3] M. Chaperon Géométrie différentielle et singularités de systèmes dynamiques, Astérisque (1986), pp. 138-139

[4] J.P. Dufour Hyperbolic actions of $R^{p}$ on Poisson manifolds (P. Dazord; A. Weinstein, eds.), Symplectic Geometry, Groupoids and Integrable Systems, Math. Sci. Res. Inst. Publ., vol. 20, Springer, New York, 1989, pp. 137-150

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