Here you can find numerical simulations of periodic solutions of the N-vortex problem carried out in Octave. Red circles are point vortices with positive strength, blue circles are vortices with negative strengths.
This is an example of a choreographic solution approximately following the boundary of the domain. These kind of solutions exist for an arbitrary number of identical point vortices and in every simply connected domain with smooth enough boundary. This has been shown in a paper together with Thomas Bartsch and Qianhui Dai. The simulation in the von Neumann domain is based on the code by Tom Ashbee. The symmetry of the domain and the choice of four vortices simplify the search for suitable initial conditions a lot.
The superposition of an equilibrium solution of a system of m vortices in a domain and m rigidly rotating configurations of the whole-plane system leads (under some conditions) to periodic solutions. This statement is part of my superposition article. The easiest example is illustrated in the simulation above. It uses an explicitly computable equilibrium of the two-vortex system in the unit disc having vorticities +1 and -1, together with two rigidly rotating vortex pairs having vorticities +1/2,+1/2 (red) and -1/2,-1/2 (blue). The result is a periodic solution of the 4-vortex problem in the disc, which is not a rigidly rotating configuration.
Besides vortex pairs also other relative equilibrium solutions of the whole-plane system can be used for the superposition with fixed equilibria to induce periodic solutions in domains. Some examples are:
Three vortices form a rigidly rotating configuration when placed as an equilateral triangle. Here the vorticities are +10,+10,-30.
The Thomson N-Gon is a (choreographic) configuration consisting of N identical vortices.
When placed at the roots of the Nth Hermitian polynomial N identical vortices rotate in a straight line.