On nonlinear stability of the regular vortex systems on a sphere

We present the necessary and sufficient conditions for stability and instability of the stationary rotation of a system of n identical point vortices located at the same latitude on a sphere at vertices of a regular n-gon. We also examine stability of the equilibrium configuration of identical point vortices, situated at the vertices of a regular polyhedra. It is proved that vortex tetrahedron, octahedron, and icosahedron are stable, while vortex cube and dodecahedron are unstable.