Phase change between separatrix crossings in slow–fast Hamiltonian systems

We consider a Hamiltonian system with slow and fast motions, one degree of freedom corresponding to fast motion, and the other degrees of freedom corresponding to slow motion. Suppose that at frozen values of the slow variables there is a non-degenerate saddle point and a separatrix on the phase plane of the fast variables. In the process of variation of the slow variables, the projection of a phase trajectory onto the phase plane of the fast variables may repeatedly cross the separatrix. These crossings are described by the crossing parameter called the pseudo-phase. We obtain an asymptotic formula for the pseudo-phase dependence on the initial conditions, and calculate the change of the pseudo-phase between two subsequent separatrix crossings.