=
(v_1, v_2, g)$ across a surface $S$ spanning the streamline of the unperturbed system $\Gm_{I_1,
I_2}$. The quantity $\Ffi$ is an integral of the averaged system (\[vp2\]) and it is an adiabatic invariant of the exact system (\[vp1\]) (see e.g. [@VVN:1996a; @NeishtadtandVasiliev:1999]). If $\dot{\th}$ does not vanish anywhere on a streamline, $\Ffi(I_1, I_2)$ would be conserved with the accuracy of order $\eps$ over times of order $1/\eps$ (see c). Note, that the conservation of $\Ffi$ is a direct consequence of the validity of averaging.

Structure of the resonance and the breakdown of the method of averaging.
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It is widely known in the theory of dynamical systems that passages through the regions where the frequencies of a given system satisfy the rational condition (in terms of (\[vp1\]), $n_1 \eps
v_1 + n_2 \eps v_2 + n_3 (\om + \eps g) = 0$, where $n_i$’s are integer numbers) lead to chaotic dynamics. As the ratio of characteristic frequencies in \[vp1\]) is of order of $\eps$, for the resonance to be of a low order we must have $\om = 0$. This is the only important resonance in the systems similar to \[vp1\]). In the process of motion some streamlines pass 