that there is only one oscillatory domain. The separatrix $\Sg$ is denoted by the dashed line. In the opposite case, the phase portrait looks like the one shown in Fig. 1b.

\[f1\]

Scattering on resonance.
------------------------

The value of $\Ffi$ changes in the process of motion near and inside the resonant zone. The cumulative change in $\Ffi$ during a single passage can be considered as a jump associated with a resonance crossing. One can see that the theory of scattering in volume-preserving systems is the same as in Hamiltonian systems developed in [@Neishtadt:1999]. We refer the reader to that paper (see also [@Vainchtein:2004]) for the details of derivation and the error estimates.

Consider a streamline that crosses the resonant surface at some time moment $t_*$. Along this streamline a phase point passes through the resonant zone fast enough, in time of order $1/\sqrt{\eps}$. While the phase point moves inside the resonant zone changes of the slow variables are small, of order $\sqrt{\eps}$. Therefore, for approximate description of dynamics inside the resonant zone we can fix the value of slow variables at their values at the crossing.

Let $t_1$ and $t_2$ be two time moments, $t_1 <t_* < t_2$ such that 