带相位滞后的离散时间Kuramoto模型：线性稳定性分析与同步起始

中文摘要 AI 辅助

我们研究了带相位滞后的Kuramoto模型的离散时间版本，该模型包含在非线性映射下演化的全局耦合的分布频率的相位振子。在无限振子数（$N\to \infty$）的连续极限下，我们推导了单振子概率密度时间演化的精确Frobenius-Perron方程，并研究了非相干态的线性稳定性。不稳定性标志着同步的开始。对于振子频率的Lorentzian分布情形，我们解析地获得了相应的同步阈值。该阈值不同于连续时间Kuramoto模型的阈值，反映了离散时间映射与连续时间流在稳定性条件上的根本差异。在同步阈值之外，我们观察到若干有趣的非线性现象：与经典Kuramoto模型不同，离散时间版本表现出周期态和混沌态。有限$N$系统的数值模拟证实了同步阈值的解析预测，同时凸显了著名的Ott-Antonsen假设的失效，该假设曾被方便地用于通过低维描述研究连续时间Kuramoto模型。

英文摘要

We investigate the discrete-time version of the Kuramoto model with phase lag, which comprises globally-coupled phase oscillators of distributed frequencies that are evolving under a nonlinear map. In the continuum limit of an infinite number of oscillators ($N\to \infty$), we derive the exact Frobenius-Perron equation for the time evolution of the single-oscillator probability density, and study linear stability of the incoherent state. Instability signals onset of synchronization. The corresponding synchronization threshold is obtained analytically for the case of a Lorentzian distribution of the oscillator frequencies. The threshold differs from that of the continuous-time Kuramoto model, reflecting the fundamentally different stability conditions for discrete-time maps and continuous-time flows. Beyond synchronization threshold, we observe several interesting nonlinear phenomena: Unlike the classical Kuramoto model, the discrete-time version exhibits periodic and chaotic states. Numerical simulations of the finite-$N$ system confirm the analytical prediction for the synchronization threshold, while highlighting breakdown of the celebrated Ott-Antonsen ansatz invoked to conveniently study the continuous-time Kuramoto model in terms of a low-dimensional description.

发表机构

• Ramniranjan Jhunjhunwala College of Arts, Science and Commerce（拉姆尼兰詹·朱恩朱纳瓦拉文理学院）
• Tata Institute of Fundamental Research（塔塔基础研究所）

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