Choreographies in the $n$-vortex problem

TL;DR

This paper investigates the existence and properties of choreographies in the $n$-vortex problem across different geometries, identifying periodic solutions and their relation to Lyapunov families through numerical continuation methods.

Contribution

It introduces a numerical approach to find Lyapunov families of periodic orbits from polygonal equilibria and links these to vortex choreographies in various settings.

Findings

Identified dense sets of Lyapunov orbits with diophantine frequency relations.

Connected periodic orbits to choreographies in the $n$-vortex problem.

Provided numerical results for multiple configurations and geometries.

Abstract

We consider the equations of motion of $n$ vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, corresponds to choreographies of the $n$ vortices. We include numerical results for all cases, for various values of $n$, and we provide key details on the computational approach.