Bifurcation of Four Liouville Tori in One Generalized Integrable Model of Vortex Dynamics

Resumo

A generalized mathematical model of the dynamics of two point vortices in the Bose–Einstein condensate enclosed in a harmonic trap and the dynamics of two point vortices in an ideal fluid bounded by a circular region are considered. In the case of a positive vortex pair, which is of interest for physical experimental applications, a new bifurcation diagram is obtained for which the bifurcation of four tori into one is shown. The presence of the bifurcations of three and four tori in the integrable model of vortex dynamics with positive intensities evidences the complex transition and the connection of the bifurcation diagrams in both limit cases. The analytical results of this study (the bifurcation diagram, the reduction to a system with one degree of freedom, and the stability analysis) form the basis of computer simulation of the absolute dynamics of vortices in a fixed coordinate system in the case of arbitrary values of the physical parameters of the model (the intensities, the vortex interaction, etc.).