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Collective behaviors of populations of coupled oscillators have attractedmuch attention in recent years. In this paper, an order parameter approach isproposed to study the low-dimensional dynam- ical mechanism of collectivesynchronizations by adopting the star-topology of coupled oscillators as aprototype system. The order parameter equation of star-linked phase oscillatorscan be obtained in terms of the Watanabe-Strogatz transformation, Ott-Antonsenansatz, and the ensemble order parameter approach. Different solutions of theorder parameter equation correspond to diverse col- lective states, anddifferent bifurcations reveal various transitions among these collectivestates. The properties of various transitions are revealed in the star-networkmodel by using tools of nonlinear dynamics such as time reversibility analysisand linear stability analysis.

Order parameter analysis of synchronization transitions on star networks