Partial synchronization in the Kuramoto model with attractive and repulsive interactions via the Bellerophon state

We study two groups of nonidentical Kuramoto oscillators with differing frequency distributions. Coupling between the groups is repulsive, while coupling between oscillators of the same group is attractive. This asymmetry of interactions leads to an interesting synchronization behavior. For small coupling strength, the mean-fields of both groups resemble a more weakly coupled Kuramoto model. After increasing the coupling strength beyond a threshold, they reach the Bellerophon state of multiple clusters of averagely entrained oscillators. A further increase in the coupling strength then asymptotically transitions from the Bellerophon state to a single synchronized cluster. During this transition, the order parameters of both groups increase and resemble an equally strongly coupled Kuramoto model. Our analysis is based on the Ott-Antonsen mean-field theory.