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Collective behaviors of coupled oscillators have attracted much attention. Inthis Letter, we propose an ensemble order parameter(EOP) equation that enablesus to grasp the essential low-dimensional dynamical mechanism of the explosivesynchronization in heterogeneous networks. Dif- ferent solutions of the EOPequation build correspondences with diverse collective states, and differentbifurcations reveal various transitions among these collective states. Thestructural relationship between the incoherent state and synchronous stateleads to different routes of transitions to synchronization, either continuousor discontinuous. The explosive synchronization is determined by the bistablestate where the measure of each state and the critical points are obtainedanalytically by using the EOP equation. Our method and results hold forheterogeneous networks with star graph motifs such as scale-free networks, andhence, provide an effective approach in understanding the routes tosynchronization in more general complex networks.

Explosive or Continuous: Incoherent state determines the route to synchronization