Exponential synchronization of hybrid coupled networks with delayed coupling.

This paper investigates exponential synchronization of coupled networks with hybrid coupling, which is composed of constant coupling and discrete-delay coupling. There is only one transmittal delay in the delayed coupling. The fact is that in the signal transmission process, the time delay affects only the variable that is being transmitted from one system to another, then it makes sense to assume that there is only one single delay contributing to the dynamics. Some sufficient conditions for synchronization are derived based on Lyapunov functional and linear matrix inequality (LMI). In particular, the coupling matrix may be asymmetric or nondiagonal. Moreover, the transmittal delay can be different from the one in the isolated system. A distinctive feature of this work is that the synchronized state will vary in comparison with the conventional synchronized solution. Especially, the degree of the nodes and the inner delayed coupling matrix heavily influence the synchronized state. Finally, a chaotic neural network is used as the node in two regular networks to show the effectiveness of the proposed criteria.