Choreographies in the \(n\)-vortex problem (Q1739543)

The authors present a systematic numerical approach to the equations of motion of $n$-vortices of equal circulation in the plane, in a disk and on a sphere. Their method is based on the symmetries of the Lyapunov families of periodic orbits that bifurcate from the polygonal relative equilibrium in a rotating frame. Highly accurate boundary value continuation techniques used with adaptive meshes to compute the Lyaponov orbits. A choreography of the $n$-polygon in the plane is invariant under scaling, because the equations are homogeneous. However, for a disk and a sphere, the $n$-vortex problem is not homogeneous, and therefore in these cases the authors determine a continuum of choreographies with fixed resonance ratio, where the choreographies are not related by scaling. Some representative orbits are given and discussed.