Normal forms with exponentially small remainder: application to homoclinic connections for the reversible $ {0}^{2+}\mathrm{i}\omega $ resonance

Gérard Iooss; Eric Lombardi

doi:10.1016/j.crma.2004.10.002

Ordinary Differential Equations/Dynamical Systems

Normal forms with exponentially small remainder: application to homoclinic connections for the reversible

0^{2 +} i ω

resonance

Presented by: Gérard Iooss

Gérard Iooss ¹; Eric Lombardi ²

¹ IUF, Institut non linéaire de Nice, UMR 6618, 1361, routes des Lucioles, 06560 Valbonne, France
² Institut Fourier, UMR 5582, université de Grenoble 1, BP 74, 38402 Saint-Martin d'Hères cedex 2, France

Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 831-838.

Abstract
Résumé

In this Note we explain how the normal form theorem already established (Iooss and Lombardi, J. Differential Equations, in press) for analytic vector fields with a semi-simple linearization enables to prove the existence of homoclinic connections to exponentially small periodic orbits for reversible analytic vector fields admitting a $0^{2 +} i ω$ resonance where the linearization is precisely not semi simple.

Dans cette Note on explique comment le théorème de formes normales avec reste exponentiellement petit déjà obtenu (Iooss et Lombardi, J. Differential Equations, à paraître) pour les champs de vecteurs analytiques ayant un linéarisé semi-simple peut être utilisé pour montrer l'existence d'orbites homoclines à des solutions périodiques exponentiellement petites pour les champs de vecteurs analytiques, réversibles au voisinage d'une résonance $0^{2 +} i ω$ où le linéarisé n'est précisément pas semi simple.

Received: 2004-09-29
Accepted: 2004-10-05
Published online: 2004-11-18

DOI: 10.1016/j.crma.2004.10.002

Author's affiliations:

Gérard Iooss ¹; Eric Lombardi ²

¹ IUF, Institut non linéaire de Nice, UMR 6618, 1361, routes des Lucioles, 06560 Valbonne, France
² Institut Fourier, UMR 5582, université de Grenoble 1, BP 74, 38402 Saint-Martin d'Hères cedex 2, France

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     author = {G\'erard Iooss and Eric Lombardi},
     title = {Normal forms with exponentially small remainder: application to homoclinic connections for the reversible $ {0}^{2+}\mathrm{i}\omega $ resonance},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {831--838},
     publisher = {Elsevier},
     volume = {339},
     number = {12},
     year = {2004},
     doi = {10.1016/j.crma.2004.10.002},
     language = {en},
}

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Gérard Iooss; Eric Lombardi. Normal forms with exponentially small remainder: application to homoclinic connections for the reversible $ {0}^{2+}\mathrm{i}\omega $ resonance. Comptes Rendus. Mathématique, Volume 339 (2004) no. 12, pp. 831-838. doi : 10.1016/j.crma.2004.10.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.10.002/

References
Cited by

[1] G. Iooss; M. Adelmeyer Topics in Bifurcation Theory and Applications, Adv. Ser. Nonlinear Dynam., vol. 3, World Scientific, 1992

[2] G. Iooss, E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields, J. Differential Equations, in press

[3] E. Lombardi Oscillatory Integrals and Phenomena Beyond all Algebraic Orders. With Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Math., vol. 1741, Springer-Verlag, Berlin, 2000

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