Jalan Sarika, Amritkar R E, Hu Chin-Kun
Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Jul;72(1 Pt 2):016211. doi: 10.1103/PhysRevE.72.016211. Epub 2005 Jul 19.
We study the synchronization of coupled maps on a variety of networks including regular one- and two-dimensional networks, scale-free networks, small world networks, tree networks, and random networks. For small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases. When nodes show synchronized behavior, we observe two interesting phenomena. First, there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently. Second, we identify two different ways of cluster formation, namely self-organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings. The synchronized clusters may be of dominant self-organized type, dominant driven type, or mixed type depending on the type of network and the parameters of the dynamics. We define different states of the coupled dynamics by considering the number and type of synchronized clusters. For the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant (epsilon) and the logistic map parameter (mu). For large coupling strengths and nonlinear coupling we find that the scale-free networks and the Caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity. For most of our study we use the number of connections of the order of the number of nodes. As the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase.
我们研究了在各种网络上耦合映射的同步情况，这些网络包括规则的一维和二维网络、无标度网络、小世界网络、树状网络以及随机网络。对于较小的耦合强度，节点呈现出湍流行为，但随着耦合增加会形成相位同步簇。当节点表现出同步行为时，我们观察到两个有趣的现象。首先，存在一些浮动类型的节点，它们在附着于某些簇和独立演化之间表现出间歇性行为。其次，我们识别出两种不同的簇形成方式，即主要具有簇内耦合的自组织簇和主要具有簇间耦合的驱动簇。根据网络类型和动力学参数，同步簇可能是主导自组织类型、主导驱动类型或混合类型。我们通过考虑同步簇的数量和类型来定义耦合动力学的不同状态。对于由逻辑斯谛映射控制的局部动力学，我们研究了耦合常数（ε）和逻辑斯谛映射参数（μ）平面上的相图。对于较大的耦合强度和非线性耦合，我们发现无标度网络和凯莱树网络比具有相同平均连通性的其他类型网络能导致更好的簇形成。在我们的大多数研究中，我们使用与节点数量量级相当的连接数。随着连接数增加，形成簇的节点数量和簇的大小通常会增加。
