AI 中文总结
本文针对具有高斯频率分布的相位振子群体，构建了基于微扰理论的近似低维约化理论，其适用于一阶及多谐波耦合，经理论分析与数值模拟验证了有效性。
AI 中文摘要
低维约化理论（如Ott-Antonsen假设）在耦合振子群体研究中发挥了关键作用，但其应用大多局限于具有有理函数形式频率分布的系统，如柯西分布。对于此类分布，留数定理可将全局序参量的动力学以有限个极点闭合，从而得到有限维常微分方程组。然而，有理频率分布通常具有重尾特性，仅存在有限个定义明确的矩，因此作为频率分布未必总是符合实际。本文针对具有高斯频率分布的弱异质性相位振子群体，基于微扰理论构建了近似低维约化理论。该理论不仅适用于Ott-Antonsen假设可应用的一阶谐波耦合情况，还适用于具有多谐波耦合的相位振子群体。理论分析与数值模拟均验证了所提出的低维约化方法的有效性。
英文摘要
Low-dimensional reduction theories such as the Ott-Antonsen ansatz have played a crucial role in the study of populations of coupled oscillators. Their application, however, has largely been restricted to systems with frequency distributions of rational-function form, such as the Cauchy distribution. For such distributions, the residue theorem allows the dynamics of the global order parameters to be closed in terms of a finite number of poles, thereby yielding a finite-dimensional system of ordinary differential equations. Rational frequency distributions, however, generally have heavy tails and only finitely many well-defined moments, and therefore may not always be realistic as frequency distributions. In this paper, we develop an approximate low-dimensional reduction theory based on perturbation theory for weakly heterogeneous populations of phase oscillators with a Gaussian frequency distribution. We construct the theory not only for first-harmonic coupling, for which the Ott-Antonsen ansatz applies, but also for populations of phase oscillators with multi-harmonic coupling. The effectiveness of the proposed low-dimensional reductions is demonstrated through both theoretical analysis and numerical simulations.
Comments11 pages, 3 figures,
