Symmetry-based selection rules for higher-order interactions in coupled oscillators

Pairwise interactions among general nonlinear oscillators can be reduced, via phase reduction, to a Kuramoto-type phase coupling $\sin(- \theta_j+\theta_k )$. For higher-order interactions, multiple phase couplings exist -- such as $\sin(-2\theta_j+\theta_k+\theta_l )$ and $\sin(-\theta_j+2\theta_k-\theta_l)$. Since different nonpairwise coupling functions produce qualitatively different dynamics, it is important to understand which phase couplings should be included in coupled phase oscillator models. In this Letter, we establish selection rules for higher-order phase coupling functions. These selection rules, which can be applied without the need of explicit phase reduction, are solely based on the symmetry of the isolated oscillator velocity field and the $n$-body interaction functions. As phase reduction established the mechanistic basis for the Kuramoto model, our results provide a theoretical link between physical systems and higher-order phase models.