Collective dynamics of higher-order coupled phase oscillators

The Kuramoto model consisting of large ensembles of coupled phase oscillators serves as an illustrative paradigm for studying the synchronization transitions and collective behaviors in various self-sustained systems. In recent years, the research of the high-order coupled phase oscillators has attracted extensive interest for the high-order coupled structure playing an essential role in modeling the dynamics of code and data storage. By studying the effects of high-order coupling, we extend the Kuramoto model of high-order structure by considering the correlations between frequency and coupling, which reflects the intrinsic properties of heterogeneity of interactions between oscillators. Several novel dynamic phenomena occur in the model, including clustering, extensive multistability, explosive synchronization and oscillatory state. The universal form of the critical coupling strength characterizing the transition from disorder to order is obtained via an analysis of the stability of the incoherent state. Furthermore, we present the self-consistent approach and find the multi-cluster with their expressions of order parameters. The stability analysis of multi-cluster is performed in the subspace getting stability condition together with the stable solutions of order parameters. The discussion of all the results summarizes the mechanism of the transition from hysteresis to oscillatory states. In addition, we emphasize that the combination of the Kuramoto order parameter capturing the asymmetry of the system and the Daido order parameter representing the clustering can give a complete description of the macroscopic dynamics of the system. The research of this paper can improve the understanding of the effects of the heterogeneity among populations and the explosive synchronization occurring in higher-order coupled phase oscillators.