Singularly perturbed Hamiltonian systems: the dynamics near slow manifold∗

When studying a slow-fast (singularly perturbed) Hamiltonian system in two degrees of freedom one can say unexpectedly much about its dynamics near its slow manifold. We demonstrate some details of this picture assuming the phase portrait of the slow system be known and generic. Since this latter system is in one degree of freedom, the problem in question is about the structure near the annulus on the slow manifold filled with periodic orbits, near a center singular point, and near a separatrix loop. In all these case we present the details of the orbit structure.