Phase portraits and the bifurcation set for the three-vortex interaction system
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.
Fields of Study & Subjects
Fluid Dynamics and Turbulent Flows
Meteorological Phenomena and Simulations
Publication details
- Journal / source
- Communications in Nonlinear Science and Numerical Simulation
- Publisher
- Elsevier BV
- Volume
- 163
- Pages
- 110428–110428
- Publication date
- 20 June 2026
- Language
- EN