Dynamical Systems And Complex Networks: Theory And Applications

In this thesis, rigorous mathematical analysis is presented for the dynamics of some models of complex networks. At first, a class of general network models: neural networks, are investigated. For both delayed and undelayed neural network systems, the criteria based on linear matrix inequality are obtained guaranteeing global stability. With these criteria, one can judge whether a neural network is globally stable; furthermore, the role of the time delay on the stability is also derived. As for the case with discontinuous activiation functions, the dynamics is quite different. The concept of the solution, viability of the solution, existence and stability of the equilibrium are thoroughly studied. When the parameters are all periodic, two effective methodologies are utilized to discuss the periodic dynamics. These methodologies are based on different understandings of the periodic trajectory. Secondly, the synchronization of two general classes of models describing complex networks: linearly coupled ordinary differential equations and linearly coupled map lattices, are studied. Based on geometrical analysis of synchronization manifold, criteria for local and global synchronization are obtained. With these criteria, one can verify whether a coupled system can be synchronized. Furthermore, the role that the topological structure of complex networks plays on synchronization can be quantitated. A quanitity is introduced to measure the synchrnizability of complex networks. For different types of coupling configuration, especially complex network models, such as random graph, small-world networks, and scale-free networks, the synchronizability is investigated theoretically and numerically. The case with delayed coupling is also considered. The influence of the coupling delay on synchronization is carefully studied. Criteria dependent on and independent on the coupling delay guaranteeing synchronization are also derived. At last, the model named by linearly coupled neural networks are introduced to apply our theoretical results to praction. Synchronous autowaves can be generated by this coupled model, with which parallel image processing and secure communication can be realized. A few real-world examples are given to indicate the applications of the synchronization of the coupled system.