Dynamics of the Restricted \((N+1)\)-Vortex Problem with a Regular Polygon Distribution

Abstract

The restricted \((N+1)\)-vortex problem is investigated in the plane with the first N identical vortices forming a relative equilibrium configuration of a regular N-polygon and the vorticity of the last vortex being zero. We characterize the global dynamics using the method of qualitative theory. It can be shown that the equilibrium points of the system are located at the vertices of three different regular N-polygons and the origin. The equilibrium points on one regular polygon are stable, whereas those on the other two regular polygons are unstable. The origin and singularities are also stable and surrounded by dense periodic orbits. For \(N=3\) or 4, there exist homoclinic and heteroclinic orbits; while for \(N\ge 5\), the system’s orbits consist of equilibrium points, heteroclinic orbits, and periodic orbits. We numerically study the trajectories of the passive tracer (a particle with zero vorticity) under specific circumstances, which support our theoretical results.