of $\rho$ does not change (see (\[vpc4\])), in the first approximation, the chaotic domain occupies the annulus between $\rho_{min}$ and $\rho_{max}$, that are the minimum and maximum values of $\rho_R(z)$, respectively, and are given by $$\rho_{min} = \sqrt{\frac1{\eta} \frac{1+\eta+\eta\dt}{1+\eta+\dt}} \; \mbox{and} \; \rho_{max} =
\sqrt{\frac1{\eta} \frac{1+\eta-\eta\dt}{1+\eta-\dt}}.$$ For the above values of the parameters, we get that the mixing domain is located between \_[min]{} = 2 \_[max]{} = 2.65. \[vpc4c\] Note, that $\rho_{min}$ and $\rho_{max}$ do not depend on the strength of perturbation, $\eps$. The division of the flow domain on the chaotic and regular subdomains is shown in Fig. 2. The parts to the left and to the right of the corresponding vertical lines are filled with KAM tori. One can see that for $\dt=1$, we get $\rho_{max} = 1/\eta$ (i.e. $r_{max} = r_o$), in other words, all the outside part of the annulus is chaotic. To mix the part adjacent to the inner cylinder, it is more effective to modulate the frequency of the inner, not of the outer, cylinder. In the opposite limit, as $\dt \to 0$, the chaotic domain shrinks to the zero width around $\rho = \sqrt{1/\eta}$.

\[f2\]

Resonance variables.
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We can define a new 