Homotopy invariants methods in the global dynamics of strongly damped wave equation
On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$
June 2016, 36(6): 3153-3225. doi: 10.3934/dcds.2016.36.3153
Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems
Tiphaine Jézéquel 1, , Patrick Bernard 2, and Eric Lombardi 3,
IUT de Lannion, Rue Edouard Branly, BP 30219, 22302 Lannion cedex, France
CEREMADE, UMR CNRS 7534 et PSL, Université Paris - Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France
Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 9, France
Received September 2014 Revised September 2015 Published December 2015
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.
Keywords: Gevrey, KAM, exponentially small phenomena, Liapunoff theorem., invariant manifolds, Hamiltonian systems, generalized solitary waves, homoclinic orbits with several loops, Normal forms, $0^2 i \omega$.
Mathematics Subject Classification: Primary: 37J20; Secondary: 37J40, 37J4.
Citation: Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153
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Tiphaine Jézéquel Patrick Bernard Eric Lombardi