Choreographies in the $n$-vortex problem
Abstract
We consider the equations of motion of 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 vortices. We include numerical results for all cases, for various values of , and we provide key details on the computational approach.
Cite
@article{arxiv.1807.08212,
title = {Choreographies in the $n$-vortex problem},
author = {Renato Calleja and Eusebius Doedel and Carlos García-Azpeitia},
journal= {arXiv preprint arXiv:1807.08212},
year = {2018}
}