Higher-order synchronization on the sphere

We construct a system of $N$ interacting particles on the unit sphere $S^{d-1}$ in $d$-dimensional space, which has $d$-body interactions only. The equations have a gradient formulation derived from a rotationally-invariant potential of a determinantal form summed over all nodes, with antisymmetric coefficients. For $d=3$, for example, all trajectories lie on the 2-sphere and the potential is constructed from the triple scalar product summed over all oriented 2-simplices. We investigate the cases $d=3,4,5$ in detail, and find that the system synchronizes from generic initial values, for both positive and negative coupling coefficients, to a static final configuration in which the particles lie equally spaced on $S^{d-1}$. Completely synchronized configurations also exist, but are unstable under the $d$-body interactions. We compare the relative effect of 2-body and $d$-body forces by adding the well-studied 2-body interactions to the potential, and find that higher-order interactions enhance the synchronization of the system, specifically, synchronization to a final configuration consisting of equally spaced particles occurs for all $d$-body and 2-body coupling constants of any sign, unless the attractive 2-body forces are sufficiently strong relative to the $d$-body forces. In this case the system completely synchronizes as the 2-body coupling constant increases through a positive critical value, with either a continuous transition for $d=3$, or discontinuously for $d=5$. Synchronization also occurs if the nodes have distributed natural frequencies of oscillation, provided that the frequencies are not too large in amplitude, even in the presence of repulsive 2-body interactions which by themselves would result in asynchronous behaviour.