Dynamical Analysis Of 3D Chaotic Systems With Different Equilibria

Since Lorenz found the first mathematical and physical model of chaos in 1963, chaos has gained a lot of attention from various fields and obtained rapid development. As a kind of complex dynamical phenomenon in nonlinear science, chaos widely exists in nature. In recent years, new breakthroughs have been emerging constantly in research of chaos theory and its practical application, which play a vital role in the development of many fields, such as secure communications, biological science and engineering.The paper proposes two kinds of new three-dimensional autonomous chaotic systems and investigates deeply the complex dynamic properties of the systems. By using the center manifold theory and normal form, the paper investigates the local dynamics such as the number, stability and bifurcation of equilibrium. Meanwhile, using the numerical simulation and Poincare compactification method, the paper also discusses the global dynamics of the system including the existence and coexisting phenomenon of chaotic attractor, periodic attractor and the dynamic behavior at infinity. The specific works of the paper are as follows:In the first chapter, we present the research background and significance of the paper and the current research status and trends of chaos theory. Then we introduce simply some basic concepts the paper will use. Meanwhile, we list several classical three-dimensional chaotic systems including Lorenz system and Chen system.In the second chapter, a new autonomous three-dimensional chaotic system with only one stable equilibrium is proposed. Firstly, the stability and bifurcation of the equi-librium are discussed by the center manifold theory and the Hopf Bifurcation theory. With the help of the tools such as the Lyapunov spectrum, the bifurcation diagram and the Poincare map, the global dynamic behavior of the system is analyzed. Under the appropriate parameters condition, the system can produce the coexisting phenomenon between chaotic attractor and periodic attractor as the initial value changes. Further, the dynamic behavior of the system at infinity is obtained by using the Poincare com-pactification method.In the third chapter, a kind of new autonomous three-dimensional chaotic system with arbitrary equilibria are studied. Firstly, the number and corresponding stability of the equilibria of the system are analyzed. By using the numerical simulation method, the dynamic properties of the system with no equilibrium and finite equilibrium are discussed in detail under the condition of proper parameters. For no equilibrium, the system can produce chaos and periodic attractor by choosing different parameters. For finite equilibrium, the chaotic systems with one, two, three, five and one hundred and eighty-seven equilibria are obtained. We further investigate the stability of corresponding equilibria.In the forth chapter, two kinds of chaotic systems with infinite isolated equilibria are analyzed. Using the center manifold theory, we study the stability of the nonhyperbolic equilibria of corresponding systems. The global dynamic behaviors of the systems are investigated by the Lyapunov spectrum, the Poincare map and the bifurcation diagram. By choosing proper parameters, on one hand, the system can produce chaos, one period and two periods; On the other hand, there exists period-doubling bifurcations to chaos.