论文标题
嘈杂的振荡器人群及其与Ott-Antonsen和Watanabe-Strogatz理论的链接的确切有限维度减少
Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories
论文作者
论文摘要
动力学方程在热力学限制中描述了全球耦合相位振荡器的种群，分布密度或等效地是由无限层次结构的等级参数的无限层次结构。 Ott和Antonsen [Chaos 18，037113（2008）]发现了一个不变的有限维子空间，每个人群的一个复合变量描述了动力学。对于具有cauchy分布频率的振荡器或对Cauchy White噪声驱动的振荡器，该子空间弱稳定，因此描述了渐近动力学。在这里，我们报告了Ott-Antonsen子空间之外的动力学的确切有限维度减少。我们表明，从通用初始状态的演变可以降低到三个复杂变量的演变，以及一个恒定的函数。对于相同的无噪声振荡器，此还原对应于渡边 - 史特罗盖茨方程系统[Phys。莱特牧师。 70，2391（1993）]。我们讨论如何使用还原系统来探索扰动合奏的瞬态动力学。
Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities, or equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and thus describes the asymptotic dynamics. Here we report on an exact finite-dimensional reduction of the dynamics outside of the Ott-Antonsen subspace. We show, that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe-Strogatz system of equations [Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles.
