Abstract
Abrupt and continuous spontaneous emergence of collective synchronization of coupled oscillators have attracted much attention. In this paper, we propose a dynamical ensemble order parameter equation that enables us to grasp the essential low-dimensional dynamical mechanism of synchronization in networks of coupled oscillators. Different solutions of the dynamical ensemble order parameter 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 the 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 dynamical ensemble order parameter 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.
Highlights
It is our motivation in this paper to reveal the mechanism of synchronization transition, especially the explosive synchronization in networks with a star motif by analyzing in a low-dimensional complex ensemble order parameter space in terms of the Ott-Antonsen method
Some typical incoherent states include the splay state defined by r < 1 with a fixed phase Φ, the in-phase state defined by r = 1 with a periodic phase Φ (t) which means the phase of all the leaves are equal with a drifting hub and the neutral state defined by time-periodic r(t) and Φ (t)
In this paper we proposed the dynamical ensemble order parameter equation in terms of the Ott-Antonsen approach to study the synchronization of coupled oscillators on a star graph
Summary
A star motif with a central hub is a typical topology in grasping the essential property of the heterogeneous networks. Where 1 ≤ j ≤ K, θh, θj are phases of the hub and leaf nodes, respectively, λ is the coupling strength. Where Δ ω = ωh − ω is the frequency difference between the hub and leaf nodes. The synchronous state is defined as φi(t) = φj(t) ≡ φ(t) and φ (t) = 0, which can be solved from Eq [3] as sin φ = Δω/(K + 1)λ. Since sin φ ≤ 1, the synchronous state exists when λ ≥ λc = Δω/(K + 1). The synchronous state is found to be stable when λ ≥ λ c by using linear-stability analysis. The upper limit critical coupling strengths respectively, where of λ f c is denoted by λcf
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