Synchronization Of Networks With Time Delays

Complex networks exist in nature and society widely, which is defined as a large set of interacting dynamical nodes connected by links. For example, the social network, an individual represents a node, the relationship between two people can be regard as edge. In the neural network, we can treat the neural cells as nodes and the neural fibers as edges. Each node is a fundamental unit having specific contents and the exhibiting dynamical behavior is affected not only by its own condition, but also by the interacting nodes. Synchronization, as a typical form that describes the collective motion and the phenomenon of network structure which leads to the emergence of the network, is one of the most significant dynamic behaviors in complex networks. In real life situations time delays play an important role to synchronization, which is usually associated with finite propagation velocities of information signals, latency times of neuronal excitations, etc. Main works of this paper are as follows:Firstly, we investigate the issue of synchronization in complex dynamical networks with system and coupling time-varying delays. The impulsive control scheme is employed for guaranteeing synchronization of the system. Based on the Lyapunov stability theory and the improved comparison theory, the sufficient synchronization conditions are derived theoretically. What's more, the result can be applied to the network without delay or only existing system delay or coupling delay.Secondly, we focus on the issue of mean square cluster synchronization in directed networks, which consist of non-identical nodes perturbed by communication noise and there exist both delay and non-delay coupling. In addition, all nodes state in coupling processes are nonlinear. The pinning control method is employed in designing controllers for guaranteeing cluster synchronization, meanwhile, all the controllers are supposed to occur with different probabilities by introducing Bernoulli stochastic variables. Some sufficient mean square synchronization conditions are derived and proved theoretically based on the Lyapunov stability theorem and stochastic analysis theory.Finally, we study the global exponential synchronization of nonlinearly coupled complex networks with delayed and non-delayed couplings and impulsive effects. Based on the pinning control scheme, only one node is pinned by state feedback controller. By employing impulsive delay differential inequality and under proper control gain, sufficient conditions are derived to guarantee the realization of the synchronization.In this paper, numerical simulations demonstrate the validity of theoretical results.