Study On Problems Of Dynamic Properties For Several Complex Dynamical Networks

Researchers, coming from different fields of science and engineering, focus on the general characteristics of the actual networks which have huge nodes and complicated connecting structure, particularly the relationship between network topology and its function, its network dynamical behavior, its formation mechanism of the structure and function and its evolution, etc. Synchronization, as a collective behavior, is one of the most interesting consequences of interactions of dynamical systems over complex networks, and a great number of research results have been achieved. In this dissertation, we synthetically apply nonlinear system theory, control theory and matrix theory to the main properties such as stability and synchronization of different kinds of complex networks. By studying the dynamic properties of these complex dynamical networks, on the one hand, we can better understand and explain the dynamical properties presented in real-word networks, and on the other hand, we can build more general network models reflecting the real-word networks and apply these theoretical results to practical applications for better performance.The main points and originalities in this thesis summarized as follow:1. Stabilization for a controlled complex dynamical network model with similarity.Real-world networks usually demonstrate similarity among nodes or on various levels, and we set up a complex dynamical network with similarity.Based on this model, we then investigate its stabilization problem. The decentralized state feedback and output feedback controllers with holographic-structure that asymptotically stabilize the network are designed respectively based on Lyapunov stability theorem and decentralized control method and such controllers are composed of a number of sub-controllers, each sub-controller possesses all the structural information of the others, that is, all sub-controllers share the same structure except for the different transformation parameters. The proposed controllers in this dissertation are relatively easier to implement in engineering in that they not only possess the decentralized property but also the holographic property.2. Synchronization analysis for a complex dynamical network with time-varying coupling delay.The present researcheson delayed complex dynamical networks focus on the cases that the coupling delays are constants.Generally, the derived functions of the time-varying delays are assumed to be bounded when dealing with the time-varying delay coupling.Obviously, such results are relative conservative to the cases of the time-varying delay coupling, especially when the time-varying delays vary very fast. In addition, the results gotten are associated with the Lyapunov-Krasovskii stability theorem instead of the Lyapunov-Razumikhin stability theorem. One fact not to neglect is that the Lyapunov-Razumikhin stability theorem is convenient to operate and effective to the fast time-varying delay, and the present thesis investigates synchronization dynamics of a complex dynamical network with fast time-varying delay based on the theorem of free-weighting matrices method. As a comparison in contrast, the thesis also investigates the same problem based on the Lyapunov-Krasovskii stability theorem but the fast coupling delay may not be required. Local and global delay-dependent synchronization criteria for the delayed complex dynamical network are derived respectively.3. Controlled synchronization for uncertain time-varying complex dynamical networks.Although complex dynamical networks are ubiquitous in many fields of science and engineering, many real-world networks are very complex in their topological structure, so it is difficult to use precise mathematical models to describe them. At the same time, networks are associated with many uncertainties, and the control theory may be indispensable when investigating their dynamic properties. The controlled synchronization has attracted increasing attentions due to its importance both in theory and practical applications. The thesis gives a controlled uncertain time-varying complex dynamical network model, and we design both decentralized state feedback and output feedback controllers to stabilize the network locally and globallywhen the nonlinear coupling terms are bounded by known nonlinear functions. In addition, we deduce several locally and globally decentralized adaptive synchronization criteria respectivelywhen the nonlinear coupling terms are bounded by known and unknown high-order polynomials.