噪声振子群的精确有限维约化及其与E-Antonsen和Watanabe-Strogatz理论的联系
Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories
摘要：
全局耦合相位振子的种群在热力学极限下由分布密度的动力学方程描述，或者等价地，由序参量的无穷级数方程描述。Ott和Antonsen [Chaos 18，037113（2008）]发现了一个不变的有限维子空间，在这个子空间上，每个种群的动力学都由一个复变量描述。对于具有柯西分布频率的振子或由柯西白噪声驱动的振子，这个子空间是弱稳定的，因此描述了渐近动力学。在这里，我们报告一个精确的有限维减少的动力学的E-Antonsen子空间之外。我们表明，从一般的初始状态的演变可以减少到三个复杂的变量，加上一个常数函数。对于相同的无噪声振荡器，这种减少对应于Watanabe-Strogatz方程系统[Phys. Rev. Lett. 70，2391（1993）]。我们讨论了如何减少系统可以用来探索扰动合奏的瞬态动力学。
英文摘要：
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.
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