Stepwise transition to higher degrees of coherence in a random network of phase oscillators

We consider a model system of phase oscillators which are connected in a random network. The network favors the connection of oscillators with close values of phases. We extend the order parameter used in the study of synchronization of phase oscillators and define generalized order parameters for the model system. We investigate the equilibrium properties of the model and reveal a phenomenon of stepwise transitions to higher degrees of coherence as the system goes through a series of second-order phase transitions for the order parameters. We also discuss a possible realization of the model in real physical systems.

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