Research On Projective Synchronization Of Complex Networks With Time-varying Delay Coupling

Complex systems are ubiquitous in nature,and these systems can be regarded as complex networks.Synchronization is one of the most extensive and typical behaviors in complex network dynamic behavior,which has important theoretical and practical significance for the study of complex network synchronization.Based on Lyapunov stability theorem,fractional calculus theory and inequality transformation theory,this paper studies the synchronization problem of complex networks with sector nonlinear input,time-varying delay coupling,external interference and other factors.The main conclusions are as follows:(1)Combination projective synchronization of integer order complex networks with time-varying delay coupling and external interference is studied.The form of combination projective synchronization is defined and the combination projective synchronization among multiple driving and response systems is transformed into the problem of system error stability.Based on Lyapunov stability theory and inequality theory,an adaptive controller is designed to synchronize two driving systems and a response system according to given scale factors.There is no delay term in the controller,so the effect of time varying delay is overcome effectively.Numerical simulation verifies the correctness of the theory analysis.(2)The combination projective synchronization of fractional-order complex networks with multiple time-varying delays coupled and external interference is studied.Compared with integer complex networks,fractional-order complex networks can simulate the dynamic behavior of systems more accurately.Therefore,the study of integer complex network synchronization is extended to fractional-order complex network synchronization.Firstly,the mathematical model of fractional-order complex networks with multiple time varying delays coupled and external interference is presented.Then,based on Lyapunov stability theory and fractional calculus theory,an adaptive controller is designed to realize the combination projective synchronization of multiple drive systems and response systems.Finally,the correctness of the theory analysis is verified by numerical simulation.(3)The function projective synchronization of interger order complex networks with multiple time-varying delay coupling and sector nonlinear input is studied.Function projective synchronization extends the scale factor from constant to function,which is a more generalized form of synchronization.Based on Lyapunov stability theory and inequality theory,an adaptive controller is designed to make the drive and response systems realize synchronization according to the given function proportion factor.Theoretical analysis shows that the proposed control scheme can effectively solve the problem that the control input is limited to the sector range.The complex network synchronization can be achieved with both under known and unknown sector boundary values.Numerical simulation verifies the effectiveness of the control scheme.(4)The function projective synchronization of fractional-order complex networks with multiple time-varying delay coupling and sector nonlinear inputs is studied.Firstly,the complex network model is extended from integer order to fractional order,and the mathematical model of fractional order complex network with multiple time-varying delay coupling and sector nonlinear input is constructed.Then,based on Lyapunov stability theory and fractional calculus theory,the known and unknown sector boundary values are considered respectively.The corresponding adaptive controller is designed to realize the function projective synchronization of the drive and response system.Finally,the effectiveness of the control scheme is verified by numerical simulation.