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The OA ansatz has attracted much attention recently, infinite-dimensionalKuramoto model could collapses to a two-dimensional system of orderdifferential equations with it. In this paper, we propose the ensemble orderparameter (EOP) equations to describe the dynamics for networks with a finitesize. To verify the effectiveness of this method, we apply it into the starnetwork and star-connected network. In the star network, numerous phasetransitions among different synchronous states are observed, three processes ofsynchronization, one process of de-synchronization and a group of hybrid phasetransitions, the processes of those transitions are revealed by the EOPdynamics and other nolinear tools such as time reversibility analysis andlinear stability analysis. Also in the star-connected network, the two-stepsynchronization transition is observed. The process of it is still be revealedby the similar methods in the single star network.

Ensemble order parameter equations in star network