  Collective behaviors of populations of coupled oscillators have attracted
much attention in recent years. In this paper, an order parameter approach is
proposed to study the low-dimensional dynam- ical mechanism of collective
synchronizations by adopting the star-topology of coupled oscillators as a
prototype system. The order parameter equation of star-linked phase oscillators
can be obtained in terms of the Watanabe-Strogatz transformation, Ott-Antonsen
ansatz, and the ensemble order parameter approach. Different solutions of the
order parameter equation correspond to diverse col- lective states, and
different bifurcations reveal various transitions among these collective
states. The properties of various transitions are revealed in the star-network
model by using tools of nonlinear dynamics such as time reversibility analysis
and linear stability analysis.
