THE MEAN FIELD ANALYSIS OF THE KURAMOTO MODEL ON GRAPHS I. THE MEAN FIELD EQUATION AND TRANSITION POINT FORMULAS
THE MEAN FIELD ANALYSIS OF THE KURAMOTO MODEL ON GRAPHS I. THE MEAN FIELD EQUATION AND TRANSITION POINT FORMULAS
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DOI：
10.3934/dcds.2019006
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发表时间：
2019-01-01
影响因子：
1.1
通讯作者：
Medvedev, Georgi S.
中科院分区：
文献类型：
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作者：
Chiba, Hayato;Medvedev, Georgi S.
在他关于同步的经典工作中，Kuramoto导出了在具有随机分布的固有频率的全对全耦合相位振子系综中对应于向同步转变的耦合强度的临界值的公式。我们将这一结果推广到确定性和随机图的收敛族上的一大类耦合系统。具体地说，我们确定了耦合强度(过渡点)的临界值，在这些临界值之间，非相干态是线性稳定的，否则是不稳定的。我们证明了过渡点依赖于由图的极限定义的核算子的最大正或/和最小负本征值(S)。这揭示了网络拓扑控制向图上的Kuramoto模型的同步过渡的精确机制。为了用具体的例子说明分析，我们推导了鄂尔多斯-仁义图族、小世界图族和k-最近邻图族上耦合系统的过渡点公式。作为独立兴趣的结果，我们对图上Kuramoto模型的平均场极限给出了严格的证明。在本文的第二部分[8]中，我们研究收敛图序列上的Kuramoto模型对应于同步开始的分支。
In his classical work on synchronization, Kuramoto derived the formula for the critical value of the coupling strength corresponding to the transition to synchrony in large ensembles of all-to-all coupled phase oscillators with randomly distributed intrinsic frequencies. We extend this result to a large class of coupled systems on convergent families of deterministic and random graphs. Specifically, we identify the critical values of the coupling strength (transition points), between which the incoherent state is linearly stable and is unstable otherwise. We show that the transition points depend on the largest positive or/and smallest negative eigenvalue(s) of the kernel operator defined by the graph limit. This reveals the precise mechanism, by which the network topology controls transition to synchrony in the Kuramoto model on graphs. To illustrate the analysis with concrete examples, we derive the transition point formula for the coupled systems on Erdos-Renyi, small-world, and k-nearest-neighbor families of graphs. As a result of independent interest, we provide a rigorous justification for the mean field limit for the Kuramoto model on graphs. The latter is used in the derivation of the transition point formulas.In the second part of this work [8], we study the bifurcation corresponding to the onset of synchronization in the Kuramoto model on convergent graph sequences.