Research On Synchronization Of Complex Dynamical Networks With Time Delays

Complex dynamical networks have been intensively studied over the last few years. The complex nature of complex networks has results in a series of important research problems. In particular, one of the most significant and interesting collective phenom-ena in complex dynamical networks is synchronization, which has attracted a large number of researchers’attention. As complex networks cannot synchronize by them-selves, there occur many control techniques to drive synchronization of the network such as global control, pinning control, impulsive control, adaptive control, intermittent control and so on.This dissertation studies the stability and synchronization control of delayed com-plex dynamical networks. The research achievements of this dissertation are as follows:1. Stability and synchronizability of the logistic complex networks with time de-lays are discussed. We obtain sufficient conditions for the stability and instability. The effects of intra-neural feedback delays and inter-neural feedback delays on the syn-chronization of complex systems under different topological structure are compared. We show that the synchronization of the network depends not only on the intrinsic dy-namical features (feedback function f) and inter-connection topology (characterized by the spectrum of the graph Laplacian) but also on the delays and the coupling strength. Moreover, there are two main findings:(i) the more neighbours, the easier to be syn-chronized; (ii) odd delays are easier to be synchronized than even ones. In addition, under regular networks or small world networks topological structure, the complex sys-tems with intra-neural feedback delays are easier to be synchronized than those with inter-neural feedback delays. At the same time, we also study the synchronization of the complex systems under scale-free network, random network and global coupled networks with feedback time delays.2. We investigate adaptive pinning synchronization in continuous/discrete time complex dynamical networks with constant delay. Based on the Lyapunov stability theory, pinning control method and linear matrix inequalities, several adaptive synchro-nization criteria are obtained. In continuous-time complex systems, we find that there exists a gap between the theoretical analysis and numerical simulation, which needs a further study in the future. Moreover, we find that small-world and scale-free network-s can reach synchronization by pin-controlling tremendously fewer nodes than regu- lar networks. In discrete-time complex dynamical networks, we design two kinds of pinning control methods (real-time pinning controllers and delay pinning controllers), obtain the synchronization criteria of delayed nonlinear systems with the identical or nonidentical topology structure. In addition, it is shown that complex systems which have both hybrid controllers and nonidentical topology structure can reach synchroniza-tion more quickly and easily.3. Because in some practical complex dynamical networks, there can occur time delay in a stochastic fashion, we investigates the problem of adaptive synchronization for continuous/discrete time complex dynamical networks with probabilistic time-varying delays. In continuous-time complex systems, we assume the probability distri-bution of the time-delays satisfying Bernoulli random distribution. Some new Lyapunov-Krasovskii functionals based on the secondary partitioning approach are constructed. Combining with the linear matrix inequality and reciprocal convex combination ap-proach technique, some delay-dependent pinning synchronization criterion are derived. For discrete-time complex dynamical networks, we assume there occurs two kinds of nonidentical probability delays, i.e. feedback delays and coupling delays, and at each instant, delays can take values in a given finite sets with probability distributions. By means of the Lyapunov theory, discrete-time Jensen inequality and reciprocal convex combination approach, several delay-dependent pinning synchronization criteria are de-rived. In addition, we use some numerical examples to illustrate the effectiveness of our theoretical analysis.4. When the system output is intermittent rather than continuous, intermittent con-trol is a more effective control method. We investigate adaptive pinning synchroniza-tion in complex delayed dynamical networks with time varying delays by intermittent control. Based on the Lyapunov stability theory and linear matrix inequalities (LMI), several adaptive synchronization criteria are obtained. Moreover, the synchronization criteria do not impose any restriction on the size of time delay.