  Collective behaviors of coupled oscillators have attracted much attention. In
this Letter, we propose an ensemble order parameter(EOP) equation that enables
us to grasp the essential low-dimensional dynamical mechanism of the explosive
synchronization in heterogeneous networks. Dif- ferent solutions of the EOP
equation build correspondences with diverse collective states, and different
bifurcations reveal various transitions among these collective states. The
structural relationship between the incoherent state and synchronous state
leads to different routes of transitions to synchronization, either continuous
or discontinuous. The explosive synchronization is determined by the bistable
state where the measure of each state and the critical points are obtained
analytically by using the EOP equation. Our method and results hold for
heterogeneous networks with star graph motifs such as scale-free networks, and
hence, provide an effective approach in understanding the routes to
synchronization in more general complex networks.
