Stable periodic motions in the problem on passage through a separatrix

A Hamiltonian system with one degree of freedom depending on a slowly periodically varying in time parameter is considered. For every fixed value of the parameter there are separatrices on the phase portrait of the system. When parameter is changing in time, these separatrices are pulsing slowly periodically, and phase points of the system cross them repeatedly. In numeric experiments region swept by pulsing separatrices looks like a region of chaotic motion. However, it is shown in the present paper that if the system possesses some additional symmetry (like a pendulum in a slowly varying gravitational field), then typically in the region in question there are many periodic solutions surrounded by stability islands; total measure of these islands does not vanish and does not tend to 0 as rate of changing of the parameter tends to 0.(c) 1997 American Institute of Physics.