The families of periodic orbits bifurcating from the fixed equilibria in a $48$-dimensional system

When inscribed inside a sphere of radius $R$, each of the five Platonic solids with a vortex of strength $\Gamma$ placed at each vertex gives rise to an equilibrium solution of the point vortex equations. In this paper, it will be shown how these equilibria can be used to generate families of periodic orbits on the sphere. These orbits are centered either around these equilibria or around more exotic equilibria, such as staggered ring configurations. Focussing on the cube as a generic case, four distinct families of periodic orbits made up of $24$ vortices (a $48$-dimensional system) are generated. These orbits bifurcate from the cube as each vertex is opened up with a splitting parameter $\theta$. The bifurcation from one orbit family to another is tracked by following the Floquet multipliers around the unit circle as the splitting parameter is varied.