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Renato Calleja


Calleja R., Doedel E., García-Azpeitia C.
Choreographies in the $n$-vortex Problem
2018, vol. 23, no. 5, pp.  595-612
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, 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 $n$ vortices. We include numerical results for all cases, for various values of $n$, and we provide key details on the computational approach.
Keywords: $n$-vortex problem, choreographies, continuation methods
Citation: Calleja R., Doedel E., García-Azpeitia C.,  Choreographies in the $n$-vortex Problem, Regular and Chaotic Dynamics, 2018, vol. 23, no. 5, pp. 595-612

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