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 ar e pulsing slowly periodically, and phase points of the system cross th em repeatedly. In numeric experiments region swept by pulsing separatr ices looks Like a region of chaotic motion. However, it is shown in th e prl:sent paper that if the system possesses some additional symmetry (like a pendulum in a slowly varying gravitational field), then typic ally in the region in question there are many periodic solutions surro unded by stability islands; total measure of these islands does not va nish and does not tend to 0 as rate of changing of the parameter tends to 0. (C) 1997 American Institute of Physics.