Periodic and quasiperiodic motion of point vortices

A number of generalizations and recent extensions of the Poincaré-Birkhoff fixed point theorem are presented, along with applications to some problems in vortex dynamics. The problem of four or more point vortices in a plane is analyzed in considerable detail for the case where all the vortex strengths have the same sign. This is accomplished using a combination of KAM theory and a recent version of the Poincaré-Birkhoff fixed point theorem. For example, it is proved that if the diameter of the initial system of vortices is sufficiently small, there exists an ample set of starting configurations that produce dynamics exhibiting quasiperiodic flows on invariant KAM tori, and periodic orbits of arbitrarily large period near these tori. It is also shown that analogous dynamical behavior occurs for configurations of any finite number of point vortices in a half-plane.