On the Critical Coupling for Kuramoto Oscillators
On the Critical Coupling for Kuramoto Oscillators
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DOI：
10.1137/10081530x
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发表时间：
2011-01-01
影响因子：
2.1
通讯作者：
Bullo, Francesco
中科院分区：
文献类型：
--
作者：
Doerfler, Florian;Bullo, Francesco
著名的仓本模型捕捉了生物和人造耦合振子动力系统中的各种同步现象。众所周知，振子之间存在一个临界耦合强度，在该强度下，发生从非相干到同步的相变。本文有四个贡献。首先，我们描述和区分不同的概念，在整个文献中使用的同步，并正式引入相位凝聚力的概念作为分析工具和性能指标同步。其次，我们回顾了大量的文献，提供必要的，充分的，隐式的和显式的估计的临界耦合强度在有限维和无限维的情况下，一阶和二阶仓本模型。第三，我们提出了第一个明确的必要和充分条件的临界耦合强度，以实现同步的有限维仓本模型的任意分布的自然频率。在同步条件的乘法间隙产生一个实际的稳定性结果，确定允许的初始和保证最终的相位凝聚力，以及保证渐近的顺序参数的大小。对于补充的结果，我们提供了一个统计比较我们的同步条件与文献中提出的其他条件，我们表明，我们的结果也适用于开关和平滑时变的自然频率。第四，也是最后，我们扩展我们的分析，多率仓本模型组成的二阶仓本振荡器的惯性和粘性阻尼与一阶仓本振荡器与多个时间常数。我们证明了这样的异构网络是局部拓扑共轭的一阶Kuramoto模型缩放的自然频率。最后，我们给出了多速率Kuramoto模型几乎全局相位同步和局部频率同步的充分必要条件。有趣的是，我们证明正确的同步条件不依赖于惯性系数，这与先前关于惯性效应在二阶仓本振子同步中的作用的观察相矛盾。
The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical systems of coupled oscillators. It is well known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength in the finite- and infinite-dimensional cases and for both first-order and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling strength to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. For supplementary results, we provide a statistical comparison of our synchronization condition with other conditions proposed in the literature, and we show that our results also hold for switching and smoothly time-varying natural frequencies. Fourth and finally, we extend our analysis to multirate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that such a heterogeneous network is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies. Finally, we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization in the multirate Kuramoto model. Interestingly, our provably correct synchronization conditions do not depend on the inertial coefficients which contradicts prior observations on the role of inertial effects in synchronization of second-order Kuramoto oscillators.