Dynamic Analysis Of Two Kinds Of Chaotic Autonomous System

It is known to all, dynamical system theory is an important part of nonlinear science, and it primarily researches the limit behaviors of natural phenomena with the evolution of time. After continuous development, the dynamical system has become one of the important branches of the modern mathematics. Chaos, as a special evolving form of nonlinear dynamical system, possesses a very important research value and application prospects in many fields.In this thesis, the development history of chaos, the basic characteristics of chaotic dynamical behavior and the related theories are introduced. With the use of theoretical analysis and numerical simulation methods, we positively consider two chaotic dynamical system. The main content is depicted as follows:In Chapter One, we introduce some research background, and some basic knowledge for chaos in Chapter Two. In Chapter Three we revisit a 3D autonomous chaotic system [10], which contains both the modified Lorenz system and the conjugate Chen system. First by citing two examples to show the errors and limitations for the local stability of the equilibrium point S+ obtained in this chapter, we formulate a complete determining criterion for the local stability of S+ of this system. Although the local bifurcation problem of this system, mainly for Hopf bifurcation, etc., has been studied, the invoking of incorrect proposition leads to an incorrect result for Hopf bifurcation. We then renew the study of the Hopf bifurcation of this system by utilizing the Project Method. Next we consider the global bifurcation problem for this system, mainly for the existence of homoclinic and heteroclinic orbits. Our results, one of which shows the existence of two heteroclinic orbits, not only correct and further supplement the ones obtained in the literature, but also give something new to theoretically help fully understand the occurrence of chaos.In Chapter Four, we formulate some new results for a 3D chaotic finance system. Firstly, we investigate the Hopf bifurcation problem of this system by utilizing the Project Method. Secondly, by using the Poincare compactification method for polynomial vector field in R3, a complete description for its dynamic behaviors on the sphere at infinity is presented. All theoretical results obtained are illustrated by numerical simulations.