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abstract: 'Resonance processes are common phenomena in multiscale (slow-fast) systems. In the present paper we consider capture into resonance and scattering on resonance in 3-D volume-preserving slow-fast systems. We propose a general theory of those processes and apply it to a class of viscous Taylor-Couette flows between two counter-rotating cylinders. We describe the phenomena during a single passage through resonance and show that multiple passages lead to the chaotic advection and mixing. We calculate the width of the mixing domain and estimate a characteristic time of mixing. We show that the resulting mixing can be described using a diffusion equation with a diffusion coefficient depending on the averaged effect of the passages through resonances.'
bibliography:
- 'refer1.bib'
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[**On passage through resonances in volume-preserving systems**]{}\

[D.L.Vainchtein$^{\left. 1,2 \right)}$, A.I.Neishtadt$^{\left. 2 \right)}$, and I.Mezic$^{\left. 1 \right)}$]{}\

[$^{\left. 1 \right)}$ *Department of Mechanical and Environmental Engineering\
University of California Santa Barbara, CA 93106, USA*]{}\

[$^{\left. 2 \right)}$ *Space Research Institute, ul.Profsouznaya, 84/32, Moscow, Russia, GSP-7, 117997*]{}\

Introduction.
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Appearing in a variety of settings, multiscale systems present quite a challenge for direct numerical simulations (see e.g. [@Jansson:2005]). Thus, a development of analytical tools is rather important. For a wide class of 