Adiabatic invariance and separatrix: Single separatrix crossing

Detailed numerical experiments on the dynamics and statistics of a single crossing of the separatrix of a nonlinear resonance with a time-varying amplitude are described. The results are compared with a simple approximate theory first developed by Timofeev and further improved and generalized by Tennyson and coworkers. The main attention is paid to a new, ballistic, regime of separatrix crossing in which the violation of adiabaticity is maximal. Some unsolved problems and open questions are also discussed.