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In this paper we study the dynamics near the equilibrium point of a family ofHamiltonian systems in the neighborhood of a $0^2 iw$ resonance. The existenceof a family of periodic orbits surrounding the equilibrium is well-known and weshow here the existence of homoclinic connections with several loops for everyperiodic orbit close to the origin, except the origin itself. To prove thisresult, we first show a Hamiltonian normal form theorem inspired by theElphick-Tirapegui-Brachet-Coullet-Iooss normal form. We then use a localHamiltonian normalization relying on a result of Moser. We obtain the result ofexistence of homoclinic orbits by geometrical arguments based on the lowdimension and with the aid of a KAM theorem which allows to confine the loops.The same problem was studied before for reversible non Hamiltonianvectorfields, and the splitting of the homoclinic orbits lead to exponentiallysmall terms which prevent the existence of homoclinic connections toexponentially small periodic orbits. The same phenomenon occurs here but we getround this difficulty thanks to geometric arguments specific to Hamiltoniansystems and by studying homoclinic orbits with many loops.

Homoclinic connections with many loops near a $0^2 iw$ resonant fixed point for Hamiltonian systems