Exact solutions of the abrupt synchronization transitions and extensive multistability in globally coupled phase oscillator populations

The Kuramoto model consisting of large ensembles of globally coupled phase oscillators serves as a paradigm for modelling synchronization and collective behavior in diverse self-sustained systems. As interest in the effects of higher-order interactions, we study an extension of the Kuramoto model with higher-order structure by considering the correlations between frequency and coupling. The resulting model is exactly solvable and several novel dynamical phenomena including clustering, multistability, and abrupt synchronization (desynchronization) transition emerge. We demonstrate that the extensive multiclusters corresponding to different arrangements of oscillator populations on the circle are established in a universal way that are independent of the choices of frequency distributions (heterogeneity of the ensembles). Using the partial dimensionality reduction of the Ott-Antonsen, we present a rigorous analysis of various multi-cluster states by studying their spectrum in the thermodynamic limit. In particular, we prove that a large multiplicity of the synchronous states are asymptotically stable to perturbation in the tangent space, thereby determining an explicit stability condition for their occurrences in the high-dimensional phase space.