The Global Phase Space of the Three-Vortex Interaction System

Abstract: We derive a symplectic reduction of the evolution equations for a system of three point vortices and use the reduced system to succinctly explain a kind of bifurcation diagram that has appeared in the literature in a form that was difficult to understand and interpret. Using this diagram, we enumerate and plot all the global phase-space diagrams that occur as the circulations of the three vortices are varied. The reduction proceeds in two steps: a reduction to Jacobi coordinates and a Lie-Poisson reduction. In a paper, we used a different method in the second step. This took two forms depending on a sign that arose in the calculation. The Lie-Poisson equations unify these into a single form. The Jacobi coordinate reduction fails when the total circulation vanishes. We adapt the reduction method to this case and show how it relates to the non-vanishing case.