论文标题
Kuramoto模型的渐近相锁定动力学和临界耦合强度
Asymptotic phase-locking dynamics and critical coupling strength for the Kuramoto model
论文作者
论文摘要
我们研究库拉莫托模型的渐近聚类（相锁定）动力学。为了分析库拉莫托流中的新出现渐近模式，我们引入了路径临界耦合强度，该耦合强度从部分相锁定到完整的相锁定，并为路径临界临界偶联强度提供了非平凡的上限。数值模拟表明，与固有频率的大小相比，多群体和单群体可以在库拉马托流中渐近出现。然而，尽管最近在足够大的耦合强度状态下，库拉莫托模型的完全同步在完全同步的库拉莫托模型上，库拉莫托流动的这种相锁动力学的理论和严格分析仍然缺乏完全理解。在本文中，我们提出了足够的框架，以根据初始相构型，耦合强度和固有频率进行多数集合的部分相锁和完整的相锁定。作为我们分析的副产品，我们从固有频率的直径，初始库拉莫托订单参数和系统大小$ n $方面获得了路径临界耦合强度的非平凡上限。我们还表明，其顺序参数小于$ n^{ - \ frac {1} {2}} $的相锁状态是线性不稳定的。
We study the asymptotic clustering (phase-locking) dynamics for the Kuramoto model. For the analysis of emergent asymptotic patterns in the Kuramoto flow, we introduce the pathwise critical coupling strength which yields a sharp transition from partial phase-locking to complete phase-locking, and provide nontrivial upper bounds for the pathwise critical coupling strength. Numerical simulations suggest that multi- and mono-clusters can emerge asymptotically in the Kuramoto flow depending on the relative magnitude of the coupling strength compared to the sizes of natural frequencies. However, theoretical and rigorous analysis for such phase-locking dynamics of the Kuramoto flow still lacks a complete understanding, although there were some recent progress on the complete synchronization of the Kuramoto model in a sufficiently large coupling strength regime. In this paper, we present sufficient frameworks for partial phase-locking of a majority ensemble and the complete phase-locking in terms of the initial phase configuration, coupling strength and natural frequencies. As a by-product of our analysis, we obtain nontrivial upper bounds for the pathwise critical coupling strength in terms of the diameter of natural frequencies, initial Kuramoto order parameter and the system size $N$. We also show that phase-locked states whose order parameters are less than $N^{-\frac{1}{2}}$ are linearly unstable.
