Department of Theoretical Physics, Perm State University, Bukirev Street 15, Perm 614990, Russia.
Institute of Continuous Media Mechanics, UB RAS, Academician Korolev Street 1, 614013 Perm, Russia.
Phys Rev Lett. 2018 Jun 29;120(26):264101. doi: 10.1103/PhysRevLett.120.264101.
We develop an approach for the description of the dynamics of large populations of phase oscillators based on "circular cumulants" instead of the Kuramoto-Daido order parameters. In the thermodynamic limit, these variables yield a simple representation of the Ott-Antonsen invariant solution [E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008)CHAOEH1054-150010.1063/1.2930766] and appear appropriate for constructing perturbation theory on top of the Ott-Antonsen ansatz. We employ this approach to study the impact of small intrinsic noise on the dynamics. As a result, a closed system of equations for the two leading cumulants, describing the dynamics of noisy ensembles, is derived. We exemplify the general theory by presenting the effect of noise on the Kuramoto system and on a chimera state in two symmetrically coupled populations.
我们提出了一种基于“循环累积量”而不是 Kuramoto-Daido 序参量来描述大规模相振荡器动力学的方法。在热力学极限下，这些变量给出了 Ott-Antonsen 不变解[E. Ott 和 T. M. Antonsen, Chaos 18, 037113 (2008)CHAOEH1054-150010.1063/1.2930766]的简单表示，并适合在 Ott-Antonsen 假设的基础上构建微扰理论。我们利用这种方法来研究小的固有噪声对动力学的影响。结果，推导出了一个描述噪声环境中动力学的两个主要累积量的封闭方程组。我们通过展示噪声对 Kuramoto 系统和两个对称耦合群体中的嵌合体状态的影响，对一般理论进行了例证。
