Phase transitions of coupled oscillators with mixed higher-order interactions

The exploration of the intrinsic mechanisms underlying synchronization has long held pervasive significance within the scientific community, as it offers profound insights for understanding the macroscopic self-organized dynamics on top of complex systems. Here, we investigate the generalized Kuramoto model incorporating the mixed higher-order interactions. In particular, we mathematically argue that, in the mean-field representation, the coupling scheme is equivalent to the biharmonic coupling, wherein the coupling function of phase oscillators goes beyond the setting of the single-Fourier modes. Importantly, we ascertain that the interplay between the pairwise single-harmonic coupling and the nonpairwise biharmonic coupling is responsible for the occurrence of a number of fascinating synchronized states, as well as extensive multistability and diverse phase transitions. We establish a general framework for comprehending the underlying mechanisms of the emergent phenomena involved, and our work, thus, provides profound insights for shaping the collective behaviors observed in the complex networks encoded with higher-order topologies.