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arXiv · 2504.16038

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

Atul Anurag·Roy H. Goodman
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Abstract

We derive a symplectic reduction of the evolution equations for a system of three interacting point vortices in the plane, first introducing Jacobi coordinates, then Lie-Poisson reductions, and finally reparameterizing the resulting leaves, arriving at an integrable system on a topologically nontrivial phase space surface. The reduced system is convenient for describing all aspects of three-vortex dynamics, including finite-time collapse, the calculation of relative equilibria and their stability, and scattering. We use the final simplified system to succinctly and geometrically explain a kind of bifurcation diagram that has appeared in the literature.

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Atul Anurag, Roy H. Goodman. 2025-04-22. Phase portraits and the bifurcation set for the three-vortex interaction system. https://doi.org/10.1016/j.cnsns.2026.110428

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