| Synchronization of complex networks has been a focus in many different fields such as biology, sociology and physics. Now, a great deal of progress on research on synchronization of complex networks has been achieved, however, most of which were discussed in the context of noise-free coupled oscillators. Due to the ubiquity of random noise and time delay, the diversity of synchronization types, and the complexity of network topology and node dynamics, the synchronization between two delay-coupled networks with noise perturbation is still an urgent issue. Based on the nonlinear dynamic systems and the theory and technique of stochastic differential delay equation, this dissertation further discusses the stochastic synchronization problem between time-delay coupled networks by designing effective control strategy. The main contents and conclusions are as follows:1. The effects of noise on lag synchronization of two unidirectionally coupled identical networks are investigated. Firstly, based on the stability theory of stochastic differential equation, the sufficient condition for lag synchronization is derived systematically by introducing a linear feedback controller with white-noise-based feedback gain. Then, comparing with the conventional linear feedback scheme, the lower bound of the sufficient condition for lag synchronization is decreased when the noise present in the controller. Thus, the constructive roles of noise in inducing or enhancing lag synchronization between coupled networks are revealed in some sense. Finally, two numerical examples are illustrated to verify the validity of the theoretical analysis.2. Generalized projective lag synchronization between delay-coupled networks under circumstance noise is studied. First, two delay-coupled complex networks with different node dynamics, different topological structures and different numbers of nodes are constructed. Second, according to the LaSalle-type invariance principle for stochastic differential delay equation, we rigorously prove that the generalized projective lag synchronization between the drive-response networks can be achieved almost surely, by introducing an appropriate controller. Furthermore, numerical simulation is employed to verify the validity of theoretical analysis, and the remarkable influences of the key parameters such as update gain, scale factor and the coupling delay on synchronization are revealed.3. The complete synchronization between noise-perturbed networks with unknown dynamical parameters is studied. In the real world, the dynamical parameters of network models are always unknown or partially unknown due to the various uncertain information. In this case, we further discuss the synchronization of noise-perturbed networks with unknown parameters and time-varying coupling delay. Based on the LaSalle-type invariance principle for stochastic differential delay equation, via designing an effective controller and the appropriate unknown parameter update laws, we theoretically prove that the synchronization between the drive-response networks can be achieved in a statistical sense of probability one. Finally, numerical simulation demonstrates that the control strategy is effective, and the synchronization of coupled networks can indeed achieve for any positive update gains. |
