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

TL;DR

This paper analyzes the bifurcation of periodic solutions in vortex and filament systems, starting from a polygonal equilibrium with a central vortex, using orthogonal degree theory to establish global bifurcation results.

Contribution

It introduces a novel bifurcation analysis for vortex and filament configurations with a central vortex, employing orthogonal degree to prove global bifurcation of periodic solutions.

Findings

Bifurcation of periodic solutions from polygonal equilibria

Existence of periodic traveling wave solutions in filament problems

Dependence of bifurcation on the circulation of the central vortex

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.