Homoclinic connections with many loops near a 02iw resonant fixed point for Hamiltonian systems
Jézéquel, Tiphaine; Bernard, Patrick; Lombardi, Eric (2014), Homoclinic connections with many loops near a 02iw resonant fixed point for Hamiltonian systems. https://basepub.dauphine.fr/handle/123456789/12505
TypeDocument de travail / Working paper
Series titlePreprints Ceremade
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Abstract (EN)In this paper we study the dynamics near the equilibrium point of a family ofHamiltonian systems in the neighborhood of a 02i! resonance. The existence of afamily of periodic orbits surrounding the equilibrium is well-known and we show herethe existence of homoclinic connections with several loops for every periodic orbit closeto the origin, except the origin itself. To prove this result, we rst show a Hamiltoniannormal form theorem inspired by the Elphick-Tirapegui-Brachet-Coullet-Iooss normalform. We then use a local Hamiltonian normalization relying on a result of Moser.We obtain the result of existence of homoclinic orbits by geometrical arguments basedon the low dimension and with the aid of a KAM theorem which allows to con nethe loops. The same problem was studied before for reversible non Hamiltonian vectorelds, and the splitting of the homoclinic orbits lead to exponentially small termswhich prevent the existence of homoclinic connections to exponentially small periodicorbits. The same phenomenon occurs here but we get round this di culty thanksto geometric arguments speci c to Hamiltonian systems and by studying homoclinicorbits with many loops.
Subjects / KeywordsGevrey; invariant manifolds; exponentially small phenomena; Hamiltonian systems; generalized solitary waves,; Liapuno theorem; homoclinic orbits with several loops,; KAM; Normal forms
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