We consider the Rydberg electron in a circularly polarized microwave
field, whose dynamics is described by a 2 d.o.f. Hamiltonian depending
on one parameter K>0, which is a perturbation of the standard Kepler
problem. The associated Hamiltonian system has two equilibria: L1
(center-saddle for all K) and L2 (center-center for small K and
complex-saddle otherwise). Associated to L1 there is a family of
Lyapunov periodic orbits that form a normally hyperbolic invariant
manifold (NHIM). In this talk, we compute the primary transversal
homoclinic orbits to the NHIM (and therefore the associated scattering
maps) combining Poincaré-Melnikov methods with numerical methods. It
should be noted that the transversality of these homoclinic orbits is
exponentially small in K (in analogy with the libration point L3 of the
R3BP).
This is a joint work with Mercè Ollé and Juan R. Pacha (Universitat
Politècnica de Catalunya).