Synchronization of coupled oscillators on small-world networks
The paper studies the dynamics of networks of coupled phase oscillators \[ \frac{d\theta_n}{dt} = \omega_n - K\sum_{m=1}^N A_{nm} \sin(\theta_n - \theta_m), \] where \(\theta_n\) and \(\omega_n\) are the phase and the natural frequency of oscillator \(n\), respectively. The coupling strength is controlled by the parameter \(K\) and the coupling structure of the network by the matrix \((A_{nk})_{n,k=1,\dots,N}\) with elements 1 or 0. The authors consider the Watts-Strogatz type small-world networks, which are defined as follows. First, a one-dimensional ring of \(N\) sites connected with all its \(k\) neighbors is prepared. Next, each edge is rewired with probability \(p\). In the case when the probability is not too big (about 0.01 for \(N=1000\)), the network has the properties of a small-world network. Using mainly numerical integration, the authors focus on the two types of synchronization: frequency and phase synchronization. The main result of the paper is to show that the critical coupling strength for the phase synchronization is larger than that of the frequency synchronization. Moreover, the critical coupling strengths for the both types of synchronization diverges as the structure of the network approaches the regular one (\(p\to 0\)). This behavior is shown to obey a power low.
scientific article; zbMATH DE number 5593881