# Bifurcation Diagram of One Generalized Integrable Model of Vortex Dynamics

## TL;DR

This paper analyzes a generalized integrable vortex model, revealing complex bifurcation structures and transitions, with implications for vortex dynamics in Bose-Einstein condensates and ideal fluids.

## Contribution

It derives analytical equations for bifurcation diagrams of a generalized vortex model, uncovering new bifurcation phenomena and connecting limiting cases.

## Key findings

- New bifurcation diagrams are obtained.
- Three-into-one and four-into-one tori bifurcations are observed.
- Bifurcation diagrams are analytically derived and parametrized.

## 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-into-one 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.