Probability phenomena due to separatrix crossing

The effect of a small or slow perturbation on a Hamiltonian system with one degree of freedom is considered. It is assumed that the phase portrait ("phase plane") of the unperturbed system is divided by separatrices into several regions and that under the action of the perturbations phase points can cross these separatrices. The probabilistic phenomena are described that arise due to these separatrix crossings, including the scattering of trajectories, random jumps in the values of adiabatic invariants, and adiabatic chaos. These phenomena occur both in idealized problems in classical mechanics and in real physical systems in planetary science and plasma physics contexts.