Point Vortices: Finding Periodic Orbits and their Topological Classification

Abstract

The motion of point vortices constitutes an especially simple class of solutions to Euler's equation for two dimensional, inviscid, incompressible, and irrotational fluids. In addition to their intrinsic mathematical importance, these solutions are also physically relevant. Rotating superfluid helium can support rectilinear quantized line vortices, which in certain regimes are accurately modeled by point vortices. Depending on the number of vortices, it is possible to have either regular integrable motion or chaotic motion. Thus, the point vortex model is one of the simplest and most tractable fluid models which exhibits some of the attributes of weak turbulence. The primary aim of this work is to find and classify periodic orbits, a special class of solutions to the point vortex problem. To achieve this goal, we introduce a number of algorithms: Lie transforms which ensure that the equations of motion are accurately solved; constrained optimization which reduces close return orbits to true periodic orbits; object-oriented representations of the braid group which allow for the topological comparison of periodic orbits. By applying these ideas, we accumulate a large data set of periodic orbits and their associated attributes. To render this set tractable, we introduce a topological classification scheme based on a natural decomposition of mapping classes. Finally, we consider some of the intriguing patterns which emerge in the distribution of periodic orbits in phase space. Perhaps the most enduring theme which arises from this investigation is the interplay between topology and geometry. The topological properties of a periodic orbit will often force it to have certain geometric properties.