Bifurcation diagram of one perturbed vortex dynamics problem

Abstract

This paper deals with the one perturbed vortex dynamics problem which describes the system of two point vortices in a Bose-Einstein condensate enclosed in a cylindrical trap. This system is a completely integrable Hamiltonian system with two degrees of freedom. Bifurcation diagram of momentum mapping is analytically investigated. To find the bifurcation diagram we use the method of critical subsystems developed by M.P. Kharlamov for the study of the phase topology of integrable problems of rigid body dynamics. As an application, we present an analysis of the stability of critical trajectories (i.e., the nondegenerate singularities of rank 1 of the momentum mapping) by defining a trajectory type (elliptic or hyperbolic) for each curve from the bifurcation set.