Xu Can, Sun Yuting, Gao Jian, Qiu Tian, Zheng Zhigang, Guan Shuguang
College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China.
Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China.
Sci Rep. 2016 Feb 23;6:21926. doi: 10.1038/srep21926.
Recently, the first-order synchronization transition has been studied in systems of coupled phase oscillators. In this paper, we propose a framework to investigate the synchronization in the frequency-weighted Kuramoto model with all-to-all couplings. A rigorous mean-field analysis is implemented to predict the possible steady states. Furthermore, a detailed linear stability analysis proves that the incoherent state is only neutrally stable below the synchronization threshold. Nevertheless, interestingly, the amplitude of the order parameter decays exponentially (at least for short time) in this regime, resembling the Landau damping effect in plasma physics. Moreover, the explicit expression for the critical coupling strength is determined by both the mean-field method and linear operator theory. The mechanism of bifurcation for the incoherent state near the critical point is further revealed by the amplitude expansion theory, which shows that the oscillating standing wave state could also occur in this model for certain frequency distributions. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogeneous couplings.
最近，人们对耦合相位振子系统中的一阶同步转变进行了研究。在本文中，我们提出了一个框架来研究具有全对全耦合的频率加权Kuramoto模型中的同步。我们进行了严格的平均场分析以预测可能的稳态。此外，详细的线性稳定性分析证明，在同步阈值以下，非相干态仅为中性稳定。然而，有趣的是，在这个区域中，序参量的幅度呈指数衰减（至少在短时间内），类似于等离子体物理中的朗道阻尼效应。此外，临界耦合强度的显式表达式由平均场方法和线性算子理论共同确定。振幅展开理论进一步揭示了临界点附近非相干态的分岔机制，该理论表明，对于某些频率分布，该模型中也可能出现振荡驻波态。我们的理论分析和数值结果相互一致，这有助于我们理解具有异质耦合的一般网络中的同步转变。
