Averaging method for systems with separatrix crossing

The averaging method provides a powerful tool for studying evolution in near-integrable systems. Existence of separatrices in the phase space of the underlying integrable system is an obstacle for application of standard results that justify using of averaging. We establish estimates that allow to use averaging method when the underlying integrable system is a system with one rotating phase, and the evolution leads to separatrix crossings.Comment: This is an author-created, un-copyedited version of the article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from i

Bifurcations of phase portraits of pendulum with vibrating suspension point

We consider a simple pendulum whose suspension point undergoes fast vibrations in the plane of motion of the pendulum. The averaged over the fast vibrations system is a Hamiltonian system with one degree of freedom depending on two parameters. We give complete description of bifurcations of phase portraits of this averaged system

Separatrix crossing in rotation of a body with changing geometry of masses

We consider free rotation of a body whose parts move slowly with respect to each other under the action of internal forces. This problem can be considered as a perturbation of the Euler-Poinsot problem. The dynamics has an approximate conservation law - an adiabatic invariant. This allows to describe the evolution of rotation in the adiabatic approximation. The evolution leads to an overturn in the rotation of the body: the vector of angular velocity crosses the separatrix of the Euler-Poinsot problem. This crossing leads to a quasi-random scattering in body's dynamics. We obtain formulas for probabilities of capture into different domains in the phase space at separatrix crossings.Comment: 18 pages, 5 figure

On passage through resonances in volume-preserving systems

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.Comment: 23 pages and 9 Figure