Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems

Tiphaine  Jézéquel; Patrick  Bernard; Eric  Lombardi

doi:10.3934/dcds.2016.36.3153

Article Contents

2016, Volume 36, Issue 6: 3153-3225. Doi: 10.3934/dcds.2016.36.3153

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Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems

1.
IUT de Lannion, Rue Edouard Branly, BP 30219, 22302 Lannion cedex
2.
CEREMADE, UMR CNRS 7534 et PSL, Université Paris - Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16
3.
Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 9

Received: September 2014

Revised: September 2015

Published: May 2017

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Abstract

In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a $0^2i\omega$ 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.

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Mathematics Subject Classification: Primary: 37J20; Secondary: 37J40, 37J45.

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