Choreographies in the $n$-vortex problem

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.

References in corpus (10)

• Absolute and relative choreographies in the problem of point vortices moving on a plane
• Unchained polygons and the N-body problem
• Periodic solutions of the N-vortex Hamiltonian system in planar domains
• Stability of Thomson's Configurations of Vortices on a Sphere
• Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type
• Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the $n$-body problem
• Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary
• New periodic solutions for three or four identical vortices on a plane and a sphere
• Choreographies in the Discrete Nonlinear Schrödinger Equations
• Relative periodic solutions of the n-vortex problem on the sphere

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• Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of $N$-Vortex Type