Pavel Ryabov

This article is devoted to the results of phase topology research on a generalized mathematical model, which covers such two problems as the dynamics of two point vortices enclosed in a harmonic trap in a Bose – Einstein condensate and the dynamics of two point vortices bounded by a circular region in an ideal fluid. New bifurcation diagrams are obtained and three-into-one and four-into-one tori bifurcations are observed for some values of the physical parameters of the model. The presence of such bifurcations in the integrable model of vortex dynamics with positive intensities indicates a complex transition and a connection between bifurcation diagrams in both limiting cases. In this paper, we analytically derive equations that define the parametric family of bifurcation diagrams of the generalized model, including bifurcation diagrams of the specified limiting cases. The dynamics of the bifurcation diagram in a general case is shown using its implicit parameterization. A stable bifurcation diagram, related to the problem of dynamics of two vortices bounded by a circular region in an ideal fluid, is observed for particular parameters’ values.

This paper is concerned with a system two point vortices in a Bose – Einstein condensate enclosed in a trap. The Hamiltonian form of equations of motion is presented and its Liouville integrability is shown. A bifurcation diagram is constructed, analysis of bifurcations of Liouville tori is carried out for the case of opposite-signed vortices, and the types of critical motions are identified.

We investigate the phase topology of the integrable Hamiltonian system on $e(3)$ found by V. V. Sokolov (2001) and generalizing the Kowalevski case. This generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. The relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the isoenergy manifolds of the reduced systems with two degrees of freedom are classified. The set of critical points of the momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the diagram of the momentum map and give a classification of isoenergy and isomomentum diagrams equipped with the description of regular integral manifolds and their bifurcations. We construct the Smale–Fomenko diagrams which, when considered in the enhanced space of the energy-momentum constants and the essential physical parameters, separate 25 different types of topological invariants called the Fomenko graphs. We find all marked loop molecules of rank 0 nondegenerate critical points and of rank 1 degenerate periodic trajectories. Analyzing the cross-sections of the isointegral equipped diagrams, we get a complete list of the Fomenko graphs. The marks on them producing the exact topological invariants of Fomenko–Zieschang can be found from previous investigations of two partial cases with some additions obtained from the loop molecules or by a straightforward calculation using the separation of variables.