evaluated at the resonance. For $(\pt \Psi(0,\sg)/\pt \sg)_* <0$, $P(M)=0$.

An example: Taylor-Couette Stokes flow.
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In the rest of the paper we discuss the chaotic advection and mixing in a particular example of volume-preserving system. Besides illustrating a general theory developed in the previous sections, we would like to demonstrate that the resonance phenomena can be used in real settings to create flows with coexisting mixing and non-mixing (KAM) domains with the respective size and mixing rates easily and independently manageable. As a model base problem we chose a relatively simple system – a Taylor-Couette flow between two infinite counter-rotating circular coaxial cylinders. Following [@CartwrightFeingoldandPiro:1996], we restrict our consideration to Stokes (small $Re$) flows. However, we would like to stress that the whole phenomena is structurally stable, which means that it survives most of the changes in the geometry of the flow settings and peculiarities of the flow perturbations.

In the system under consideration, the unperturbed flow is given by (a solution to a 2-D Stokes equation): &=& 0,\
&=& 0, \[vpc1\]\
&=& ( r,z), with $$\om(r,z) = \om_i \frac{r}{r_i} \left( 1- \frac{1}{1-\eta^2} \left( 1-\eta\frac{\om_o}{\om_i}
\right) \right) + \om_i \frac{r_i}{r}\left( 1- \eta\frac{\om_o}{\om_i} \right) \frac{1}{1-\eta^2},$$ where $$\eta = r_i/r_o.$$ In 