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Formes normales de champs de vecteurs : restes exponentiellement petits dans le cas non autonome périodique et orbites homoclines à plusieurs boucles au voisinage de la résonance 0²iw hamiltonienne.

Abstract : In this thesis we consider two problems dealing with normal forms of vector fields and exponentially small phenomena. In the first chapter, we prove two results of normalization with exponentially small remainders for analytic vectorfiels in the neighborhood of a fixed point, in a periodic nonautonomous case. The first normalization theorem allows to construct a quasi-invariant manifold with an exponentially small remainder while the second one is a normal form result of the Elphick-Tirapegui-Brachet-Coullet-Iooss type with an exponentially small remainder. In the second chapter, we study the dynamic near the equilibrium point of a family of hamiltonian systems in the neighborhood of a 0²iw resonance. We first show the existence of a family of periodic orbits surrounding the equilibrium and then the existence of homoclinic orbits with several loops for every periodic orbit close to the origin, except the origin itself. The proof is based on a hamiltonian normal form theorem proved in this chapter, inspired by the Elphick-Tirapegui-Brachet-Coullet-Iooss normal form and on a local hamiltonian normalization relying on a result of Moser. We obtain the result of existence of homoclinic orbits by geometrical arguments based on the low dimension and with the aid of a KAM theorem which allows to confine the loops. The same problem was studied before for reversible non hamiltonian vectorfields, and the splitting of the homoclinic orbits lead to exponentially small terms which prevent the existence of homoclinic connections to exponentially small periodic orbits. The same phenomenon occurs here but we get round this difficulty thanks to geometric arguments specific to hamiltonian systems.
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Contributor : Tiphaine Jézéquel <>
Submitted on : Wednesday, December 7, 2011 - 6:02:12 PM
Last modification on : Thursday, March 5, 2020 - 5:57:09 PM
Long-term archiving on: : Thursday, March 8, 2012 - 2:35:52 AM

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  • HAL Id : tel-00649382, version 1

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Tiphaine Jézéquel. Formes normales de champs de vecteurs : restes exponentiellement petits dans le cas non autonome périodique et orbites homoclines à plusieurs boucles au voisinage de la résonance 0²iw hamiltonienne.. Equations aux dérivées partielles [math.AP]. Université Paul Sabatier - Toulouse III, 2011. Français. ⟨tel-00649382⟩

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