Complex Dynamics Of A Lorenz-5D Hyperchaotic System

Since Lorenz first found the chaotic attractor in 1963, the theory of chaos has obtained the unprecedented development in many fields. In the recent half century, chaos has become one of the most core subjects in the nonlinear science. the Lorenz system,which has been known as the first mathematical model of chaos, is an important milestone and has great significance in the development of chaos. A chaos system with two or more positive Lyapunov exponents is known as hyperchaos system, this implies that its dynamics are expended in several di?erent directions simultaneously. Thus, hyperchaos systems have more complex dynamical behavior than chaotic systems. The randomness and unpredictability of hyperchaos systems is higher than chaotic systems, too. At the same time, hyperchaos has widely application in the theory and technological fields, such as secure communications, nonlinear circuits, neural network, etc. Recently, because of the high dimensional of hyperchaos system, the research of hyperchaos theory mainly focused on four-dimensional system. There are very few discussions in five-dimensional system, especially for the system with three positive Lyapunov exponents.Based on Lorenz-type systems, by introducing a linear and a nonlinear feedback controller to the Lorenz-type system, this paper proposes a kind of five dimensional controlled Lorenz-type system with three positive Lyapunov exponents. We strict prove that the Lorenz-5D system with a set of hyperchaotic parameters is smoothly nonequivalent to the Yang-5D hyperchaotic system in the case of the same linear controller. We also investigates the dynamic behaviors of this system. Using the center manifold theory,normal form and geometric theory of di?erential equation, this paper investigates the dynamics such as the stability of hyperbolic and nonhyperbolic equilibrium, the pitchfork and Hopf bifurcation. Meanwhile, this paper further discusses the dynamics at infinity of this hyperchaotic system. The main works of this paper are as follows:In Chapter 1, we introduce the background of this paper and the current research at homeland and abroad, including the development phase of chaos and hyperchaos theory.Meanwhile, we list the method and the theory which will be used in this paper. In this chapter, the research in typical hyperchaotic systems and the Lorenz-type hyperchaotic systems of four and five-dimensional are summarized.In Chapter 2, we propose a new class of five dimensional controlled Lorenz-type system with three positive Lyapunov exponents(hereafter referred to as Lorenz-5D system).we discuss the stability of the hyperbolic equilibrium, and give the su?cient and necessary conditions of them. When the Lorenz-5D system and the Yang-5D hyperchaotic system have the same linear controller, we prove that they are smoothly nonequivalent.Chapter 3 discusses the stability of nonhyperbolic equilibrium and the local bifurcation of the new 5D system. Using the center manifold theory and normal form, we give the stability of nonhyperbolic equilibrium with one and two zero eigenvalues. We study the pitchfork bifurcation by the parameter-dependent center manifold theory. When the bifurcation parameter equal to the critical value, we also gain the dynamic behavior at the equilibrium of the new system. On the other hand, we study the Hopf bifurcation by employing the high-dimensional Hopf bifurcation theory and applying symbolic inference.The expression and the direction of the bifurcating periodic solution is investigated, too.In Chapter 4, we discuss the global dynamic behavior of the Lorenz-5D hyperchaotic system. At first, according to the techniques of Lyapunov exponents spectrum, bifurcation diagram, Poincar′e map, we investigate the dynamic behaviors of this hyperchaotic system. When the system choose appropriate parameters, this system can exhibit hyperchaotic(especially with three positive Lyapunov exponents), chaotic, periodic and quasi periodic dynamics. When fix the value of parameters and change the initial conditions,there are several coexisting attractors in the new system, such as hyperchaotic and periodic attractors, hyperchaotic and quasi-periodic attractors, two periodic attractors. Furthermore, using the technique of high-dimensional Poincar′e compactification and the invertible linear transformation, we project the phase space of the vector field about the Lonrez-5D system onto five coordinates, and analysis the vector field in the new coordinates to gain the dynamics at infinity of Lonrez-5D system.