Adaptive synchronization of complex dynamical networks with non-delayed and delayed coupling

The paper investigates the global synchronization of nonlinearly coupled complex dynamical networks with non-delayed and delayed coupling. The delays are assumed to vary and be continuously differentiable. Adaptive controllers have been designed for nodes of the controlled networks based on Lyapunov stability theory. Analytical results show that the states of dynamical networks can globally asymptotically synchronize to a desired orbit under the designed controllers. What is more, the node dynamic need not satisfy the very strong and conservative uniformly Lipschita condition and the coupling matrix is not assumed to be symmetric or irreducible. A numerical example of undirected delayed dynamical networks is given, respectively, using the Lorenz system as the nodes of the networks, which demonstrates the effectiveness of proposed results.