low-dimensional multiscale systems (that are also called slow-fast systems) where some variables change with a characteristic rate of order $1$, while the other variables change with a rate of order of $\eps \ll 1$ the averaging method can be used for approximate description of the dynamics (see e.g. [@AKN:1988]). A standard condition for applicability of the averaging method is that the frequency of motion of the fast variables is indeed of order $1$ everywhere in the space. The violation of the above condition near resonances of the fast system may give rise to the following [*single resonance phenomena*]{}: first, separatrix crossings or, second, scattering on resonance and capture into resonance. The hydrodynamic flows involving the separatrix crossings were introduced in [@BajerandMoffatt:1990; @StoneNadimandStrogatz:1991] and later discussed from the theoretical, numerical and experimental perspectives (see e.g. [@WardandHomsy:2001; @WardandHomsy:2003; @Grigoriev:2005]). The theory of separatrix crossings for such flows was developed in [@VVN:1996a; @NeishtadtandVasiliev:1999].

The flows where the chaotic advection is caused by scattering on resonance and capture into resonance were introduced in [@CartwrightFeingoldandPiro:1995; @CartwrightFeingoldandPiro:1996]. The later works include [@Solomon:2003]. However, the corresponding theory, that gives a quantitative description of either single crossings or the long-time dynamics was not developed.

There are two major 