Bifurcation of periodic solutions from a ring configuration in the vortex and filament problems

Abstract

This paper gives an analysis of the movement of n+1 almost parallel filaments or vortices. Starting from a polygonal equilibrium of n vortices with equal circulation and one vortex at the center of the polygon, we find bifurcation of periodic solutions. The bifurcation result makes use of the orthogonal degree in order to prove global bifurcation of periodic solutions depending on the circulation of the central vortex. In the case of the filament problem these solutions are periodic traveling waves.