Bifurcations and linear stability of families of relative equilibria with a dominant vortex

Abstract

Studying the dynamics of vortex configurations with one large vortex with and N smaller vortices has applications to physical electromagnetic systems and atmospheric science, as well as being a historically interesting problem. The existence, and linear stability of relative equilibria configurations of the (1+N)-vortex problem are examined. Such configurations are shown to be critical points of a special potential function, and their linear stability depends on the weighted Hessian of this potential. Algebraic geometry and some numerical methods are used to examine the bifurcations of critical points and stability specifically in the case of N=3.