between the perturbations that results in the chaotic advection.

The averaged system and the structure of the resonance.
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One can see that in (\[vpc3\]) the variables $\rho$ and $z$ are slow and the variable $\th$ is fast. In the first approximation we can average (\[vpc3\]) over $\th$ to get the following averaged equations of motion (cf. (\[vp2\])): &=& 0,\
\[vpc4\]\
&=& ( 1+ / ).

The averaged trajectories (in the full 3-D, $(\rho,z,\th)$, space) wind the cylinders of constant radius with the direction of the rotation depending on the value sign of $\om$. System (\[vpc4\]) is Hamiltonian with Hamiltonian = r\_i ( + 1[ ]{} ( - (- 1) ) ). \[vpc4a\] The quantity $\Ffi$ is analogous to $H$ in [@CartwrightFeingoldandPiro:1996]. It is an integral of the averaged system and is an adiabatic invariant of the exact system: in the absence of resonances it would be conserved with the accuracy of order $\eps$ over times of order $1/\eps$ (see [@Bogolyubov:1961]).

The averaging is valid away from a resonance surface [R]{}, where $\om = 0$, that is given by $$\rho_R(z) = \sqrt{\frac1{\eta} \frac{1+\eta-\eta\dt\sin(\la z)}{1+\eta-\dt\sin(\la z)}}.$$ Trajectories that do not intersect [R]{} remain regular. As in the averaged system the value 