Bifurcation Research On A Smooth Chua Chaotic System

Since the discovery of the first chaotic attractor by American meteorologist Lorenz in 1963 when studying atmospheric problems,chaos has attracted widespread attention from scientists.They found that there are chaotic "shadows" in various fields,such as the atmospheric changes caused by the vibration of butterfly wings in nature,the dynamic instability of structures in engineering technology and the mechanism of infectious diseases in biology,etc.There are chaotic footprints everywhere.As one of the most important way to the chaos,bifurcation is an important topics in the field of chaos theory research.Based on the non-smooth chaotic Chua circuit proposed by American scientist Leon Ong Chua,this paper proposes a modified smooth Chua system,and studies its zero-Hopf bifurcation,classical Hopf bifurcation and Bogdanov-Takens bifurcation.The full text is divided into five chapters.The first chapter mainly introduces the research and development process of bifurcation and chaos,describes the close relationship and research significance of bifurcation and chaos.Introduced the development of bifurcation theory,and given the Hopf bifurcation of planar system and high-dimensional system from the mathematical point of view Theorem and other theoretical knowledge;introduces the development of chaos theory,introduces common definitions of chaos from the perspective of mathematics,the characteristics of chaotic phenomena,and classical approaches to chaos.Described the background of the Chua circuit,and the important position and research situation of the Chua system in the dynamic system are introduced.In the second chapter,the nonlinear part of the classical Chua system is linearly fitted by a cubic polynomial,and a modified smooth Chua chaotic system is given.The distribution of the equilibrium points of the smooth Chua circuit system is studied.The parameter conditions for the Chua system to generate the classical Hopf bifurcation are obtained.The critical value parameters of the Hopf bifurcation generated by the system are proposed,and then the existence of the periodic solution of the smooth Chua system is obtained by analyzing the first Lyapunov coefficient.We have proved the theorem and made a numerical simulation.In the third chapter,the theory of averaging method is introduced and the zero-Hopf bifurcation are studied.By using the characteristic equations of the smooth Chua system at the equilibrium point,and the parameter conditions to generate the zero-Hopf bifurcation are obtained.By using the first-order averaging method and the second-order averaging method,the existence of the periodic solution is proved,and the judgment basis of the stability of the periodic solution is given.The fourth chapter studies the existence and nature of the Bogdanov-Takens bifurcation of the smooth Chua system.The central flow manifold theorem and the normal form theory are used to analyze the dynamic behavior near the Bogdanov-Takens bifurcation point of the system.The corresponding bifurcation map is depicted,and the complex dynamic behaviors of Hopf bifurcation,fork bifurcation and homoclinic bifurcation of the periodic orbit are found.The fifth chapter further summarizes the content of the full text,makes introductions for the areas that can be improved,analyzes and forecasts the direction and content that can be studied in the future.