The Spectrum of the Partially Locked State for the Kuramoto Model
The Spectrum of the Partially Locked State for the Kuramoto Model
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DOI：
10.1007/s00332-006-0806-x
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发表时间：
2007-02
影响因子：
3
通讯作者：
Renato Mirollo;S. Strogatz
中科院分区：
文献类型：
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作者：
Renato Mirollo;S. Strogatz
我们解决了耦合振荡器Kuramoto模型长期存在的稳定性问题。这个系统引起了数学上的注意，部分原因是它在从神经科学到凝聚态物理等领域的应用，也因为它在非线性动力学和统计力学之间提供了一个美丽的联系。该模型由大量具有全对全正弦耦合的相位振荡器组成。振子的固有频率根据规定的概率密度随机分布在总体中，这里取其平均值为单峰和对称。随着振子之间耦合的增加，系统自发同步：靠近频率分布中心的振子将其相位锁定在一起并以同一频率运行，而尾部的振子则保持不锁定并以不同频率漂移。虽然这种“部分锁定”状态已经在模拟中观察了几十年，但它的稳定性从未被数学分析过。部分困难在于如何合理地给出模型的无限n极限。在这里，我们描述了这样一个连续极限，并证明了相应的部分锁定态实际上是中性稳定的，这与人们可能期望的相反。讨论了这一结果可能产生的影响。
We solve a long-standing stability problem for the Kuramoto model of coupled oscillators. This system has attracted mathematical attention, in part because of its applications in fields ranging from neuroscience to condensed-matter physics, and also because it provides a beautiful connection between nonlinear dynamics and statistical mechanics. The model consists of a large population of phase oscillators with all-to-all sinusoidal coupling. The oscillators' intrinsic frequencies are randomly distributed across the population according to a prescribed probability density, here taken to be unimodal and symmetric about its mean. As the coupling between the oscillators is increased, the system spontaneously synchronizes: The oscillators near the center of the frequency distribution lock their phases together and run at the same frequency, while those in the tails remain unlocked and drift at different frequencies. Although this "partially locked" state has been observed in simulations for decades, its stability has never been analyzed mathematically. Part of the difficulty is in formulating a reasonable infinite-N limit of the model. Here we describe such a continuum limit, and prove that the corresponding partially locked state is, in fact, neutrally stable, contrary to what one might have expected. The possible implications of this result are discussed.