Smale's horseshoe map in a Hamiltonian system around a separatrix

A non-linear system with one degree of freedom is studied. It is assumed that the potential of the system is a double-well type and that a time-dependent external field is imposed on the system as a perturbation. The analytic solutions around a separatrix are obtained using the first order perturbation. These analytic solutions are in good agreement with the numerical integrations. It is further shown that these solutions draw a horseshoe map of the Smale type.