into resonance, upon the arrival to the resonant zone a phase point drifts for a long time (of order $\sim 1/ \eps$) along the resonant surface. As a result, $\Ffi$ changes by a value of order $1$. Among all the streamlines that arrive to the resonant zone during a given time interval of order $\sim 1/ \eps$ only a small part, of order $\sim\sqrt{\eps}$, is captured. Initial conditions for streamlines that are or are not captured are mixed. Therefore, it is reasonable to consider capture as a probabilistic phenomenon.

The streamlines that cross the resonant zone without being captured typically pass through this zone in time of order $\sim 1/\sqrt{\eps}$ (see [@Neishtadt:1997; @Neishtadt:1999] for more accurate estimates for Hamiltonian systems). The major resonant phenomenon for such streamlines is the scattering on resonance. In this case the difference, $\Dt \Ffi = \Ffi^+ - \Ffi^-$, that is typically of order $\sim\sqrt{\eps}$, is referred to as the amplitude of the scattering or, as $\Ffi$ is the adiabatic invariant of the averaged system, as a [*jump of the adiabatic invariant*]{}. This value is very sensitive to small changes of the initial conditions: a change of the initial conditions far from a resonance by a 