Synchronization And Complicated Dynamics In A Complex Network With Multiple Time Delays

Complex network synchronization is a phenomenon in which the state outputs of the system tend to be gradually identical between nodes in the network or between different parts of the network under the action of external control signals.Most of the existing control methods apply control signals to the network nodes,which usually require control gains of a certain magnitude to achieve synchronization.It is noted that the number of connected edges in the network is usually larger than the number of nodes.If the control signal is applied to the connected edges between nodes,there will be more choices to achieve the optimal control.In this paper,we study the synchronization and complex dynamical behavior of complex networks with multiple time delays.The state control and solution space of time-delayed systems is infinite dimensional,and the uncontrolled delay will make the network appear degraded and produce instability and complexity,which makes the system control difficult.In addition,the complex topology structure of the network also brings challenges to the effective control.The paper presents a synchronization problem for complex networks with time delays based on edge control.Firstly,the local stability of the equilibrium point of the controlled error system is studied,and the boundary of the stability region of the system is judged by analyzing the characteristic roots of the adjacency matrix of the system using the characteristic root method.How to select the controlled connected edges and how to determine the control gain and control delay for complete synchronization according to different values of the coupling delay are analyzed in detail.The generalized synchronization problem of complex networks,Hopf bifurcation,three types of codimension-2 bifurcation and the complex dynamical behavior near the bifurcation point are also studied.The proposed control method is applied to study the synchronization,bifurcation,and multiple stability problems of ring delayed networks,scale-free networks,and small-world networks.The results show that the proposed control method based on connected edges is an effective method for the dynamic control of complex networks.