Xu Can, Wang Xuan, Zheng Zhigang, Cai Zongkai
Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China.
Phys Rev E. 2021 Mar;103(3-1):032307. doi: 10.1103/PhysRevE.103.032307.
The Kuramoto model serves as an illustrative paradigm for studying the synchronization transitions and collective behaviors in large ensembles of coupled dynamical units. In this paper, we present a general framework for analytically capturing the stability and bifurcation of the collective dynamics in oscillator populations by extending the global coupling to depend on an arbitrary function of the Kuramoto order parameter. In this generalized Kuramoto model with rotation and reflection symmetry, we show that all steady states characterizing the long-term macroscopic dynamics can be expressed in a universal profile given by the frequency-dependent version of the Ott-Antonsen reduction, and the introduced empirical stability criterion for each steady state degenerates to a remarkably simple expression described by the self-consistent equation [Iatsenko et al., Phys. Rev. Lett. 110, 064101 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064101]. Here, we provide a detailed description of the spectrum structure in the complex plane by performing a rigorous stability analysis of various steady states in the reduced system. More importantly, we uncover that the empirical stability criterion for each steady state involved in the system is completely equivalent to its linear stability condition that is determined by the nontrivial eigenvalues (discrete spectrum) of the linearization. Our study provides a new and widely applicable approach for exploring the stability properties of collective synchronization, which we believe improves the understanding of the underlying mechanisms of phase transitions and bifurcations in coupled dynamical networks.
Kuramoto模型是研究耦合动态单元大集合中的同步转变和集体行为的一个具有启发性的范例。在本文中，我们通过将全局耦合扩展为依赖于Kuramoto序参量的任意函数，提出了一个用于解析捕捉振荡器群体中集体动力学的稳定性和分岔的通用框架。在这个具有旋转和反射对称性的广义Kuramoto模型中，我们表明，表征长期宏观动力学的所有稳态都可以用由Ott-Antonsen约化的频率相关版本给出的通用轮廓来表示，并且为每个稳态引入的经验稳定性准则退化为一个由自洽方程描述的非常简单的表达式[Iatsenko等人，《物理评论快报》110, 064101 (2013年)PRLTAO0031 - 900710.1103/PhysRevLett.110.064101]。在这里，我们通过对约化系统中各种稳态进行严格的稳定性分析，详细描述了复平面中的频谱结构。更重要的是，我们发现系统中涉及的每个稳态的经验稳定性准则与其由线性化的非平凡特征值（离散频谱）确定的线性稳定性条件完全等价。我们的研究为探索集体同步的稳定性特性提供了一种新的且广泛适用的方法，我们相信这增进了对耦合动态网络中相变和分岔潜在机制的理解。
