Dynamical Analysis Of Complex Networks And Control And Synchronization Of Chaotic Systems

In this thesis, we mainly study dynamical behavior of small-world networks and Cohen-Grossberg neural networks, and control and synchronization of some chaotic systems. It is well known that small-world networks and Cohen-Grossberg neural networks are two classes of very important models in complex networks, and have wide range of applications in many fields. So it is very valuable for dynamical analysis of the above networks from the point of view of theoretical study and applications. In addition, chaos phenomenon has been attached more importance by scientific researchers. At present, control and synchronization of chaos are two focuses of study on chaos, and have been applied successfully to many fields, such as engineering, economy, secure communication, medical treatment, and so on. The main works in this thesis are listed as follows:In Chapter 1, we introduce the research background and progress for complex networks and chaos. At the same time, we provide the structure of the thesis.In chapter 2, we investigate local stability and Hopf bifurcation influenced by delay for one dimensional and high dimensional small-world networks, respectively. And some theorems are obtained. Then, the direction and stability of bifurcating periodic solutions are investigated by normal form and center manifold theory. At last, by numerical simulations, we further illustrate the effectiveness of theorems obtained.In chapter 3, by Brouwer's fixed point theorem, we prove existence of the equilibrium point for high-order Cohen-Grossberg neural networks with delay, and investigate global stability of the equilibrium point by Lyapunovmethod. On the other hand, by constructing new auxiliary equations and employing theory of almost periodic differential equations and compressive mapping principle, existence of almost periodic solution is investigated for Cohen-Grossberg neural networks with delay and almost periodic coefficients, furthermore, sufficient condition is given to guarantee global convergence of solutions by Lyapunov method.In chapter 4, we use Lyapunov method to investigate global exponential stability and robust stability of the equilibrium point for Cohen-Grossberg neural networks with impulse. In addition, by constructing proper Lyapunov function and combining with compressive mapping principle, periodic dynamical behavior is studied for a class of Cohen-Grossberg neural networks with impulse.In chapter 5, we discuss control of a new four dimensional chaotic systems by feedback control and impulsive control methods. Moreover, based on theory of stochastic differential equations, stochastic synchronization is studied for some chaotic systems admitting noise perturbation.At the end of this dissertation, we list problems for our future work which includes complex networks and chaotic systems.