This paper investigates the pinning synchronization of nonlinearly coupled complex networks with time-varying coupling delay and time-varying delay in dynamical nodes. Some simple and useful criteria are derived by constructing an effective control scheme and adjusting automatically the adaptive coupling strengths. To validate the proposed method, numerical simulation examples are provided to verify the correctness and effectiveness of the proposed scheme.
In the past few years, the analysis and controllability of complex networks have attracted lots of attention [
For the complexity of the dynamical network, it is difficult to realize the synchronization by adding controllers to all nodes, such as [
Motivated by the above discussions, in this paper, we work on the pinning synchronization of nonlinearly coupled complex networks with time-varying coupling delay and time-varying delay in dynamical nodes. The main contributions of this paper are threefold.
The rest of this paper is organized as follows. The network model is introduced, and some necessary definitions, lemmas, and hypotheses are given in Section
The network with time-varying coupling delay and adaptive coupling strengths can be described by
Assume that
Let
Suppose that
We now introduce some definitions, assumptions, and lemmas that will be required throughout this paper.
Assuming that the real parts of the eigenvalues of
Then we have that
If
For any two vectors
Let
For the vector-valued function
Suppose that there exist nonnegative constants
In this section, a control scheme is developed to synchronize a delayed complex network with time-varying delay dynamical nodes to any smooth dynamics
For convenience in later use, we denote
Suppose that Assumptions
Construct the following Lyapunov function:
Compared with the other control methods in the literature, pinning controller is relatively simple and is easy to implement. As we know now, the real-world complex networks normally have a large number of nodes. Therefore, it is usually difficult to control a complex network by adding the controllers to all nodes. To reduce the number of the controllers, a natural approach is to control a complex network by pinning part of nodes. In this paper, we designed controllers to ensure that the special networks could get synchronization. The pinning nodes can be randomly selected. It indeed provides some new insights for the future practical engineering design.
Synchronization criteria have been given in Theorem
In this section, a numerical example will be given to demonstrate the validity of the synchronization criteria obtained in the previous sections. Considering the following network:
The problems of synchronization and pinning control for the nonlinearly coupled complex networks with time-varying coupling delay and time-varying delay in dynamical nodes are investigated. It is shown that synchronization can be realized via adjusting time-varying coupling strengths. The study showed that the use of simple control law helps to derive sufficient criteria which ensure that the synchronization of the network model is derived. In addition numerical simulations were performed to verify the effectiveness of the theoretical results. Compared with existing results, our synchronization is still very useful when the existing methods become invalid (Figures
The chaotic behavior of time-delayed Lorenz system.
Time evolution of the synchronization errors.
Research is partially supported by the National Nature Science Foundation of China (no. 70871056) and by the Six Talents Peak Foundation of Jiangsu Province. The authors are also grateful to Miss Mary Opokua Ansong, at the Department of Computer Science and Technology, Jiangsu University, for making time to go through this work.