# On bifurcation of the four Liouville tori in one generalized integrable model of the vortex dynamics

## TL;DR

This paper investigates bifurcations in a generalized integrable vortex dynamics model, revealing complex transitions and new bifurcation diagrams involving four Liouville tori, with implications for physical vortex systems.

## Contribution

It introduces a new bifurcation diagram for positive vortex pairs and analyzes the bifurcation of four tori, connecting different limit cases in vortex dynamics.

## Key findings

- New bifurcation diagram with four tori bifurcation
- Analytical reduction to a one-degree-of-freedom system
- Stability analysis of vortex configurations

## Abstract

The article deals with a generalized mathematical model of the dynamics of two point vortices in the Bose-Einstein condensate enclosed in a harmonic trap, and of the dynamics of two point vortices in an ideal fluid bounded by a circular region. 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 indicated. The presence of bifurcations of three and four tori in the integrable model of vortex dynamics with positive intensities indicates a complex transition and the connection of bifurcation diagrams of both limit cases. Analytical results of this publication (the bifurcation diagram, the reduction to a system with one degree of freedom, the stability analysis) form the basis of computer simulation of 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 and etc.)