Synchronization Of Complex Networks With Distributed Delay Via Intermittent Control

Complex network has been applied in many fields,such as biology,physics,sociology,and so on,and drawn increasing attention from researchers.Synchronization is an important and interesting collective phenomenon in network,while it is hard to be achieved without inputs,especially synchronization with a given orbit.Researchers have proposed a lot of synchronization control methods.Intermittent control as a typical discontinuous control method,for its low cost,high efficiency and easy implementation,has been widely adopted.This paper mainly studies the synchronization of complex networks with distributed delays via intermittent control.Firstly,we investigate the exponential synchronization of real-variable complex network with distributed time delays via intermittent control.Based on the mathematical analysis theory and Lyapunov function method,we present some synchronization criteria with respect to the system parameters,control gain and the control rate.From the criteria,for any given dynamical network,we can easily estimate the needed values of control gains and rate.At last,we give two numerical simulations to verify the validity of the results.Secondly,we extend the above results to the complex-variable network.This is to say,we investigate exponential synchronization in complex-variable network with distributed delays via intermittent control.Based on the Lyapunov function method and complex analysis theory,we give some synchronization criteria.In order to reduce the computational complexity,we use the matrix analysis theory to reduce the dimension of the synchronization criterion,which is easy to verify the needed values of control values.In the end,we give two numerical simulations to verify the derived theoretical results.Thirdly,we investigate cluster synchronization in community network with distributed delays.Cluster synchronization means that the nodes in the same community achieves the same goal,but those in different communities achieve the different ones.To realize cluster synchronization,we use intermittent control method to design effective controllers for different communities.Based on the Lyapunov function method and mathematical analysis theory,we present some cluster synchronization criteria.Finally,we summarize the main ideas of this paper,discuss some results about intermittent control,and put forward some issues deserving further studies.