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