Phase transition and scaling in Kuramoto model with high-order coupling

The classical Kuramoto model serves as a useful tool for studying synchronization transitions in coupled oscillators that is limited to the sinusoidal and pairwise interactions. In this paper, we extend the classical Kuramoto model to incorporate the high-order structures and non-pairwise interactions into the coupling function. Using a self-consistent approach and constructing parametric functions, we describe the extensive multi-cluster states induced by high-order structures and identify various types of phase transitions toward synchrony. In particular, we establish the universal scaling relation for each branch of multiclusters, which describes the asymptotic dependence of the order parameters (Kuramoto and Daido) on the coupling strength near the critical points.