Adaptive Modified Function Projective Synchronization for Uncertain Complex Dynamic Networks With Multiple Time-Varying Delay Couplings Under Input Nonlinearity

Abstract

This paper is concerned with the modified function projective synchronization of the uncertain complex dynamic networks with both input nonlinearity and multiple time-varying delay couplings. Firstly, the model of the complex dynamic networks with sector nonlinear input, multiple timevarying delay couplings, model uncertainty and external disturbance is constructed. Then, the adaptive controller is formulated based on the Lyapunov stability theory and the matrix inequality theory, by which the network nodes of the driving system and the response system can realize modified function projective synchronization according to the corresponding function scaling factors. Finally, a four-dimensional hyperchaotic system is considered as the nodes of the complex dynamic networks to achieve numerical simulation. The corresponding theoretical proof and computer simulation are worked out to demonstrate the effectiveness and feasibility of the proposed scheme. INDEX TERMS Complex dynamic networks, multiple time-varying delay couplings, input nonlinearity, modified function projection synchronization, adaptive control.

Function Projective Synchronization of Two Complex Networks with Unknown Sector Nonlinear Input and Multiple Time-Varying Delay Couplings

“…Lemma 8 (see [40] Theorem 9. If Assumptions 2-5 are satisfied, the drive system (1) and the response system (2) can realize function projective synchronization with the controller (7) and adaptive laws (8)- (12).…”

“…Although the researches on sector nonlinear input have achieved certain results, most existing studies mainly focus on a single system rather than complex networks. Recently, Fang et al [40] studied the modified function projective synchronization of complex dynamic networks with sector nonlinear input. In the controller design, it is assumed that the range of the sector nonlinear input is known.…”

“…However, it is difficult to determine the exact boundary value of the sector nonlinear input. Once the restricted boundary of the control input is unknown, the controller designed in [40] is no longer applicable. How to realize function projective synchronization of complex dynamic networks under unknown sector nonlinear input is a challenging research topic.…”

Lag exponential synchronization of fractional-order complex-valued dynamical networks with multiple delays via adaptive control

“…Synchronization is one of the important collective phenomena in complex networks [9][10], which refers to the process that the motion state of network nodes tends to be identical as time goes on. There are many types of synchronization, for instance, fixed time synchronization, quasi synchronization, and projection synchronization [11][12][13]. The drive and response systems of above synchronization are started at the same time.…”