Elizaveta Artemova

In this paper, we consider the dynamics of two interacting point vortex rings in a Bose – Einstein condensate. The existence of an invariant manifold corresponding to vortex rings is proved. Equations of motion on this invariant manifold are obtained for an arbitrary number of rings from an arbitrary number of vortices. A detailed analysis is made of the case of two vortex rings each of which consists of two point vortices where all vortices have same topological charge. For this case, partial solutions are found and a complete bifurcation analysis is carried out. It is shown that, depending on the parameters of the Bose – Einstein condensate, there are three different types of bifurcation diagrams. For each type, typical phase portraits are presented.

Describing the phenomena of the surrounding world is an interesting task that has long attracted the attention of scientists. However, even in seemingly simple phenomena, complex dynamics can be revealed. In particular, leaves on the surface of various bodies of water exhibit complex behavior. This paper addresses an idealized description of the mentioned phenomenon. Namely, the problem of the plane-parallel motion of an unbalanced circular disk moving in a stream of simple structure created by a point source (sink) is considered. Note that using point sources, it is possible to approximately simulate the work of skimmers used for cleaning swimming pools. Equations of coupled motion of the unbalanced circular disk and the point source are derived. It is shown that in the case of a fixed-position source of constant intensity the equations of motion of the disk are Hamiltonian. In addition, in the case of a balanced circular disk the equations of motion are integrable. A bifurcation analysis of the integrable case is carried out. Using a scattering map, it is shown that the equations of motion of the unbalanced disk are nonintegrable. The nonintegrability found here can explain the complex motion of leaves in surface streams of bodies of water.

The motion of a spherical robot with periodically changing moments of inertia, internal rotors and a displaced center of mass is considered. It is shown that, under some restrictions on the displacement of the center of mass, the system of interest features chaotic dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the case of fixed rotors and periodically changing moments of inertia.

This paper is concerned with the problem of the interaction of vortex lattices, which is equivalent to the problem of the motion of point vortices on a torus. It is shown that the dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate configurations are found and their stability is investigated. For two vortex lattices it is also shown that, in absolute space, vortices move along closed trajectories except for the case of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero total strength are considered. For three vortices, a reduction to the level set of first integrals is performed. The nonintegrability of this problem is numerically shown. It is demonstrated that the equations of motion of four vortices on a torus admit an invariant manifold which corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices on this invariant manifold and on a fixed level set of first integrals are obtained and their nonintegrability is numerically proved.