On Aref's vortex motions with a symmetry center (Q1080720)

The paper studies point vortex motion with discrete symmetry of rotation. The main geometrical technique is the method of symplectic reduction which enables the authors to study the phase portrait of the system and gain very specific information about it. Bifurcations are detected numerically by using the ratio of vortices as the bifurcation parameter. It is proved that steady rotations exist for any number of rings. For the case of two rings, two conjectures of Aref are proved. One deals with the ratio of equilibria of two rings of opposite circulations as the number of vortices increases and the other addresses the non-integrability of the three ring problem; the latter is proved with the aid of the Melnikov method. The paper is very well written and consists of a nice blend of symplectic geometry, topology, fluid dynamics, and numerical bifurcation experiments.