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arXiv · 1807.08212

Choreographies in the $n$-vortex problem

Renato Calleja·Eusebius Doedel·Carlos García-Azpeitia
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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.

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Renato Calleja, Eusebius Doedel, Carlos García-Azpeitia. 2018-07-21. Choreographies in the $n$-vortex problem. https://doi.org/10.1134/s156035471805009x

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