Abstract
This paper studied the cluster synchronization of directed complex networks with time delays. It is different from undirected networks, the coupling configuration matrix of directed networks cannot be assumed as symmetric or irreducible. In order to achieve cluster synchronization, this paper uses an adaptive controller on each node and an adaptive feedback strategy on the nodes which in-degree is zero. Numerical example is provided to show the effectiveness of main theory. This method is also effective when the number of clusters is unknown. Thus, it can be used in the community recognizing of directed complex networks.
Highlights
During last decade, the study of complex networks has become a hot topic in various fields like physics, mathematics, biology, social sciences, computer sciences, and so on [1,2,3]
Cluster synchronization is a middle state of the progress which is from none-synchronization to complete synchronization
Owing to the significant application in biological science and communication engineering, the researching of cluster synchronization is focus on the control method such as pinning control, adaptive control, impulsive control, and so on, but few of them studies cluster synchronization of directed complex networks with time delays
Summary
The study of complex networks has become a hot topic in various fields like physics, mathematics, biology, social sciences, computer sciences, and so on [1,2,3]. There are some papers on cluster synchronization of directed networks without time delays. Ma and others intensively studied the pinning cluster synchronization of directed complex networks in [11] They gave the pinning controllers which are applied to inter-act nodes and intra-act nodes with zero in-degree, respectively. This paper uses an adaptive controller to make a directed network with time delays achieved cluster synchronization. When the complex network achieves cluster synchronization, for any node i,j, the following equation is established. The complex network can be considered to achieve P-cluster synchronization when the following condition is satisfied:. Under assumption 1 and 2, the controlled complex network Eq (3) with adaptive controller Eq (4) can achieve cluster synchronization if k is satisfied with the following equation. In order to make complex network Eq (1) achieved cluster synchronization, an adaptive controller ui(t) is added on each node. ~ eiT 1⁄2fi(xi(t),xi(t{t1)){fi(svi (t),svi (t{t1))z aijxj(t{t2){ aijsvj (t{t2){kei
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