Homoclinic orbits with many loops near a 02iω resonant fixed point of Hamiltonian systems
Jézéquel, Tiphaine; Bernard, Patrick; Lombardi, Eric (2016), Homoclinic orbits with many loops near a 02iω resonant fixed point of Hamiltonian systems, Discrete and Continuous Dynamical Systems. Series A, 36, 6, p. 3153-3225. 10.3934/dcds.2016.36.xx
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-01251087
Journal nameDiscrete and Continuous Dynamical Systems. Series A
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Institut de Recherche Mathématique de Rennes [IRMAR]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Institut de Mathématiques de Toulouse UMR5219 [IMT]
Abstract (EN)In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a 02iω resonance. The existence of a family of periodic orbits surrounding the equilibrium is well-known and we show here the existence of homoclinic connections with several loops for every periodic orbit close to the origin, except the origin itself. The same problem was studied before for reversible non Hamiltonian vector fields, and the splitting of the homoclinic orbits lead to exponentially small terms which prevent the existence of homoclinic connections with one loop to exponentially small periodic orbits. The same phenomenon occurs here but we get round this difficulty thanks to geometric arguments specific to Hamiltonian systems and by studying homoclinic orbits with many loops.
Subjects / KeywordsNormal forms; exponentially small phenomena; invariant manifolds; Gevrey; 02iω; Hamiltonian systems; homoclinic orbits with several loops; generalized solitary waves; KAM, Liapunoff theorem
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