Phase portraits and the bifurcation set for the three-vortex interaction system

We derive a symplectic reduction of the evolution equations for a system of three interacting point vortices in the plane. We first introduce Jacobi coordinates, then perform Lie–Poisson reduction, and finally reparameterize the resulting symplectic leaves. This leads to an integrable system on a topologically nontrivial reduced phase-space surface.

The reduced formulation provides a convenient framework for describing finite-time collapse, relative equilibria and their stability, scattering, and the global organization of three-vortex dynamics. We use this simplified geometric system to explain succinctly a class of bifurcation diagrams that has previously appeared in the literature.