Complex Dynamics On The Network Model And Random Effects

Complex networks have been considered as an important approach for describing and understanding complex systems by many researchers recently. Any complex system is composed of interacted individuals, which can be naturally represented by graphs with individuals denoted by nodes and interactions by links. From this point of view, complex networks are ubiquitous, ranging from nature and biological system to society. The well-known networks include neural networks, Internet, epidemic spreading networks, protein-protein interaction networks, collaboration networks, etc. To understand the structures and properties of networks, the study of dynamical behaviors on these complex networks are needed firstly. By studying the dynamic properties of complex networks, one can not only understand the dynamic properties presented in real-world networks but also take different measures for networks with different structures. Furthermore, these results can been applied to design real networks to achieve some desirable that benefit all over the world.The spread of epidemics, computer virus and rumors can be seen as same behaviors with some laws. In the traditional spreading models, the individuals are considered to be mixed adequately, that means each individual contacts other individuals with the same rate. However the recent study of complex networks shows that the distribution of individuals is not homogeneous, but has some special structures. These special structures make the spreading behavior on complex networks different from the traditional models. In order to make the model more practical, one must consider the special structures in studying the spreading behavior. Based on the work of many researchers this dissertation extend the previous results on complex networks, which are the spreading model with non-linear incidence rate and with two interacting species. Furthermore, stochastic factor will affect the dynamic behaviors of complex networks, so the effect of the stochastic noise on complex network, stochastic discrete-time neural networks, and the synchronization of discrete-time systems with common noise are also investigated. These results are significant not only in mathematical theory but also in many applied fields.The main work in this dissertation is listed as follows:In chapter 1, the research background and progress on complex networks, including neural networks, are introduced. Moreover, some preliminaries and the structure of dissertation are given.In chapter 2, the spreading models on complex networks with a generalized nonlinear incidence rate are presented. Firstly the model on homogeneous networks with nonlinear incidence rate is considered, and the existence, the stability of equilibria and the Hopf bifurcation of the model are given. Then the model on a heterogenous scale-free network are considered, and the stability of the disease-free equilibrium is obtained. It is shown that the basic reproductive number is independent of the functional form of the nonlinear incidence rate, while the number of the equilibria and the behaviors are indeed different from the corresponding model with linear incidence rate.In chapter 3, models for the spread of two interacting species on complex networks are presented. The dynamic behaviors of the models on the homogeneous network and heterogenous scale-free network are considered, and the stability of the disease-free equilibrium is obtained. Three immunization strategies are applied to models. The analytical and simulated results are given to show that the proportional immunization strategy is effective on heterogenous scale-free networks.In chapter 4, the general stochastic nonlinear models of spreading, describing the effect of random fluctuations on complex networks are proposed. It has been found that fluctuation noise would trigger a state of networks from instability to stability. The probability density function for the proportion of infected individuals are found explicitly, and the stochastic bifurcation is analyzed by probability density function. It is a better explanation of occurring the different phenomena with sensitive parameters in many real-world complex networks.In chapter 5, the effect of stochastic noise about the discrete-time stochastic neural networks are investigated. By using the martingale convergence theorem, the almost surely stability condition of two sub-classed of stochastic neural networks are analyzed. Furthermore, numerical examples are provided to illustrate some possible applications of the theoretical results.In chapter 6, the synchronization of discrete-time system and the discrete-time system with delay are discussed. These systems are all induced by common external noise. A set of sufficient conditions for the synchronization is found. These conditions show that the synchronization occurs in a wide class of discrete-time system and the discrete-time system with delay.At the end of this dissertation, the conclusions and some topics for future work which include complex networks and stochastic dynamical systems are given.