Synchronization for two delayed bi-directionally coupled networks with noise perturbation via adaptive couplings

Noise perturbation and time delay are ubiquitous in both natural and man-made systems, and must be taken into consideration in the design of synchronization. For solving the synchronization problem between two delayed bi-directionally coupled networks with noise perturbation, a synchronization criteria based on adaptive coupling was firstly presented in this paper. Based on LaSalle-type invariance principle for stochastic differential delay equations, the adaptive update laws of the coupling strengths are obtained to realize the synchronization. And the configuration matrices of the networks are not required to be symmetric and irreducible. Numerical simulations are given to show the correctness and effectiveness of theoretical results.