resonance cause $\Ffi$ diffuse and drift. The rate of drift is determined by the ensemble average of $\Dt \Ffi$ (the derivation is similar to [@Neishtadt:1999], see also [@Neishtadt:2005]): = \_0\^1 () d = - 1[2a]{} (0,) \~, \[vp7\] where $\Psi(0,\sg)$ is the flux of $f_{2,0}$ through the area (that we denote by $S_R$) inside the separatrix $\Sg$ on the $(\th, \th^{\pr})$ phase plane in Fig. 1a: (0,) = \_ f\_[2,0]{} d d\^. \[vp8\] Note, that in Hamiltonian systems (and in many volume-preserving systems, in particular in the system discussed below) $f_{2,0}$ does not depend on $\th$ and, therefore, $f_{2,0}$ can be taken out of the integral and $\Psi$ becomes proportional to $S_R$.

If the phase portrait looks like the one shown in Fig. 1b, the mean change of $\Ffi$ is zero: as there is no separatrix, $\Psi(0,\sg)=0$.

The rate of diffusion depends on the dispersion of $\Dt \Ffi$: $$D = \int_0^1 \left( \Dt \Ffi(\xi) - \left< \Dt \Ffi \right> \right)^2 d\:\xi \sim \eps.$$ We return a more detailed description of the long-time dynamics in Sect. 6.

Capture into resonance.
-----------------------

The other phenomenon that affects the behavior of streamlines at a resonance crossing is capture into resonance.

Capture is possible 