Matrix Projective Synchronization for a Class of Networks with Double Subsystems via the Dynamical Assistance of Incoming Links

The matrix projective synchronization for a class of complex dynamic networks with double subsystems and different dimensional nodes is investigated and realized via designing the dynamical tracking target of incoming link subsystem and the control input of the node subsystem in this paper. From the perspective of the large system, a complex dynamical network can be regarded as a coupling system of the node subsystem and the incoming link subsystem (double subsystems) . This kind of networks with double subsystems is the investigation aim of this paper, where the incoming weight between each couple of nodes is taken as the state component of the incoming link subsystem, and the vector differential equation is used to model the dynamical equation of the node subsystem and the incoming link subsystem, respectively. It is worth pointing out that the nodes in the network can have different state dimensions. Based on the Lyapunov stability theory and via the rigorous theoretical derivation, the auxiliary tracking target of the incoming link subsystem is designed and the control strategy of the node subsystem is proposed for the network . It can be deduced that the matrix projective synchronization of our network is sure to be realized when the incoming link subsystem has tracked the auxiliary tracking target. Finally, a proper example which embodies the characteristics of our network is given. Numerical simulations illustrate that all the matrix projective synchronization error curves of nodes tend to zero as time goes to infinity when the incoming link subsystem has tracked the auxiliary tracking target. That is to say, the matrix projective synchronization of our network is achieved via the dynamical assistance of the incoming links and by the control input on the node subsystem. This verifies the validity of the matrix projective synchronization strategy proposed in this paper.