Slow-fast hamiltonian dynamics near a ghost separatix loop

Abstract

We study the behavior of a slow-fast (singularly perturbed) Hamiltonian system with two degrees of freedom, losing one degree of freedom at the singular limit ɛ = 0, near its ghost separatrix loop, i.e., a homoclinic orbit to a saddle equilibrium of the slow (one degree of freedom) system. We show that, for small ɛ > 0, the system has an equilibrium of the saddle-center type and prove, using the method of Delatte, that the Moser normal form exists in an O(ɛ)-neighborhood of the equilibrium. Then we show that one-dimensional separatrices of the equilibria are generically split with exponentially small splitting. Also, we demonstrate that out of some exponentially thin neighborhood of the ghost separatrix loop in the level of a Hamiltonian containing a saddle-center, the major part of the phase space is foliated into Diophantine invariant tori.