Outer Synchronization Analysis And Computation Of Complex Networks

Over the past one decade, complex network behaviors have attracted a great deal of attention in variety of fields including biology, physics, mathematics, engineering and so on. As a kind of generalized form of synchronization inside a network, researches on outer synchronization of networks have the strong importance and potential applications in our life. For example, in order to know more about the communication of the infectious diseases, such as Mad Cows, AIDS and SARS, between animals and the human beings, it is required two different networks to distinguish animals from the human beings; investigating the interactions of protein network and gene network may disclose evolution process in systems biology. Based on the theory of dynamic system, algebra and control, this dissertation is to investigate the outer synchronization of networks further more. The main contents of this dissertation can be summarized as follows:1. Generalized synchronization (GS) between two coupled complex networks is theoretically and numerically studied, where the node vectors in different networks are not the same, and the numbers of nodes of both networks are not necessarily equal. The presence of GS between two systems means that there is some functional relation between the states of both systems. The chosen interaction between two networks is based on the open-plus-closed-loop method. To more accurately characterize the conditions of the occurrence of GS, a sufficient and necessary condition, i.e., the response network is uniformly asymptotically stable, is presented. And a sufficient criterion for GS of two coupled networks is established based on the auxiliary system method and the Lyapunov stability theory. Numerical examples are also included which coincide with the theoretical analysis.2. Synchronization between two symmetrical coupled populations of nonidentical phase oscillators is investigated. First, we perform a nonlinear transformation on the original model into a linear reformulation model. Then, explicit expressions of the synchronization order parameters which include the phase coherence within each population and the phase coherence between two populations can be found with some algebraic knowledge. Finally, numerical example is given to illustrate the theoretical results.3. Anti-synchronization (AS) and its control between two coupled networks are studied. AS phenomenon, which can be characterized by vanishing of the sum of relevant state variables, is a noticeable phenomenon in chaotic systems. First, AS of two coupled networks with both equivalent and different topological structures are investigated. Based on the adaptive control method and Barbalat Lemma, we introduce two simple control schemes to achieve AS, respectively. Next, AS of two different networks is studied. Here we focus on the nonlinear signal's connection and bidirectional actions, and adaptive controllers are designed to anti-synchronize them. Furthermore, we investigate the AS problem of two complex dynamical networks with non-delayed and delayed coupling using pinning adaptive control method. Based on Schur complement and Barbalat Lemma, a sufficient condition is derived to guarantee the AS between two networks. Numerical simulations are also presented to show the effectiveness of the proposed AS criterion.