Deformation of geometry and bifurcation of vortex rings

TL;DR

This paper develops a framework for analyzing vortex dynamics on curved surfaces, revealing new bifurcation behaviors and stability results for vortex configurations without traditional normal form methods.

Contribution

It introduces a smooth family of Hamiltonian systems with symmetries for point vortices on surfaces, enabling bifurcation analysis and stability conclusions without Birkhoff normal form.

Findings

Characterizes equivariant bifurcations in vortex systems.

Establishes stability of the Thomson heptagon.

Provides a new approach to vortex stability analysis.

Abstract

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.