Hopf Bifurcation And Synchronization Of Lorenz-like Chaotic System

With the popularity of computer application and the continuous development of nonlinear science in recent decades, the bifurcation and chaos theory, a complex non-linear dynamics behavior, not only has a great development in applied mathematics, physicsand mechanics, but also is widely applied in engineering, social science and almost all natural science. Hopf bifurcation, a kind of typical dynamic bifurcation and a bridge between determinate solution and periodic solutions, is a hot topic in research of the dynamics behavior of chaotic system. Chaos control and chaos synchronization has made a great breakthrough, and is becoming more mature, but the methods of chaotic research emerge in endlessly, that is able to motivate scholars and experts to continue to further study and research.In this paper, we mainly study hopf bifurcation of a Lorenz-like system and diff-erent methods to implement chaos synchronization, further using MATLAB for nume-rical simulation to verify the feasibility of these methods. On one hand, the existence and stability of the equilibrium is discussed, further we discuss the existence of hopf bifurcation near the equilibrium, and on this base, further discuss direction, stability and periodicity of the periodic solution of Hopf bifurcation. On the other hand, mainly study chaos synchronization of Lorenz-like system by using methods of backstepping control and adaptive control. The outline of the rest of this paper is organized as follows:In the first chapter of this paper, some background knowledge of chaotic theory and the preliminary knowledge which is going to be used in this paper is given. At the same time, we introduce the main work of this paper.In the second chapter of this paper, we mainly study the Hopf bifurcation of the lorenz-like system. Firstly, the existence and stability of the equilibrium is discussed; Secondly, the discussion of hopf bifurcation is given, including the existence and the direction,stability,eriodicity of periodic solution based on hopf bifurcation theorem. Finally, the accuracy of theoretical results is verified through the numerical simulation by the plotting function of MATLAB.In the third chapter of this paper, we mainly study chaos synchronization of the lorenz-like system based on the backstepping control method and lyapunov stability theory, further the validity and the effectiveness of backstepping control method is verified through the numerical simulation.In the fourth chapter of this paper, we mainly study chaos synchronization of the lorenz-like system based on adaptive control method and lyapunov stability theory, further the feasibility of backstepping control method is verified through numerical simulation.Finally, the work of this paper is summarized, and some prospects is made for the future research.