Reduction of the planar 4-vortex system at zero momentum

TL;DR

This paper analyzes a special 4-vortex system with specific strengths, reducing it via symmetries to simpler systems on a cylinder, revealing key dynamical features like equilibria and periodic orbits.

Contribution

It explicitly computes the reduction of the 4-vortex system at nongeneric momenta, classifying the resulting dynamics and showing they are equivalent to one-degree-of-freedom systems on a cylinder.

Findings

Reduced systems have one stable and one unstable equilibrium.

All orbits are periodic except for two homoclinic connections.

The reduced dynamics are qualitatively identical to simple one-degree-of-freedom systems.

Abstract

The system of four point vortices in the plane has relative equilibria that behave as composite particles, in the case where three of the vortices have strength $-\Gamma/3$ and one of the vortices has strength $\Gamma$. These relative equilibria occur at nongeneric momenta. The reduction of this system, at those momenta, by continuous and then discrete symmetries, classifies the 4-vortex states which have been observed as products of collisions of two such composite particles. In this article I explicitly calculate these reductions, and show they are qualitatively identical one degree of freedom systems on a cylinder. The flows on these reduced systems all have one stable equilibrium and one unstable equilibrium, and all the orbits are periodic except for two homoclinic connections to the unstable equilibrium.