Super-homoclinic orbits and multi-pulse homoclinic loops in Hamiltonian systems with discrete symmetries
1997, Volume 2, Numbers 3-4, pp. 126-138
Author(s): Shilnikov L. P., Turaev D. V.
Author(s): Shilnikov L. P., Turaev D. V.
4D-Hamiltonian systems with discrete symmetries are studied. The symmetries under consideration are such that a system possesses two invariant sub-planes which intersect each other transversally at an equilibrium state. The equilibrium state is supposed to to be of saddle type; moreover, in each invariant sub-plane there are two homoclinic loops to the saddle. We establish the existence of stable and unstable invariant manifolds for the bouquet comprised by the four homoclinic trajectories at the Hamiltonian level corresponding to the saddle. These manifolds may intersect transversely along some orbit. We call such a trajectory a super-homoclinic one. We prove that the existence of a super-homoclinic orbit implies the existence of a countable set of multi-pulse homoclinic trajectories to the saddle.
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