Bifurcation Diagram of One Generalized Integrable Model of Vortex Dynamics

Abstract

The article is devoted to the results of a phase topology research on a generalized mathematical model, which covers such two problems as dynamics of two point vortices enclosed in a harmonic trap in a Bose-Einstein condensate and dynamics of two point vortices bounded by a circular region in an ideal fluid. New bifurcation diagrams are obtained and three-intoone and four-into-one tori bifurcations are observed for some values of the model's physical parameters. 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 the equations, that define the parametric family of the generalized model's bifurcation diagrams, including bifurcation diagrams of the specified limiting cases. The dynamics of the general case's bifurcation diagram is shown, using its implicit parametrization. The 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 values of parameters.

Dynamics of a few vortices in an ideal fluid and in a Bose-Einstein condensate

“…In this paper, we define a bifurcation diagram as a two-parameter family. Figure 2 shows the new type of the bifurcation diagram from this family, which was not observed in any of the works [25], [18], [28]. Such a diagram contains two "loops" with the cusped points, and one of the loops has only one vertical asymptote.…”

“…In this paper, we derive a bifurcation diagram of the momentum map in the case of a generalized system that describes the dynamics of two vortices of opposite intensities, both in an ideal fluid placed inside a circular cylinder and in a Bose-Einstein condensate enclosed in a harmonic trap. Recall that in a case of two vortices of the same sign, considered in [18], the bifurcation diagram of vortices in a Bose-Einstein condensate differs significantly from the diagram of vortices in an ideal fluid. Here we found that, on the one hand, there are diagrams similar to those constructed in the [25], [18], [28], and on the other hand, there are also diagrams that differ significantly from those had been already encountered before.…”

“…Recall that in a case of two vortices of the same sign, considered in [18], the bifurcation diagram of vortices in a Bose-Einstein condensate differs significantly from the diagram of vortices in an ideal fluid. Here we found that, on the one hand, there are diagrams similar to those constructed in the [25], [18], [28], and on the other hand, there are also diagrams that differ significantly from those had been already encountered before.…”

“…. , 15 along the line h = h 2 = −5.7 are presented in the Table 1, where the points p 2 , p 5 , p 7 , p 4 Reduction to a system with one degree of freedom In the works [28], [18] bifurcation diagrams were obtained in the case of two identical vortices and vortices with intensities of the same signs, respectively. In the previous sections of this work, the presentation did not imply any restrictions on the signs of the vortex intensities.…”