Complex Dynamical Research Of A Modified Lorenz Type 5D Hyperchaotic System

Since Lorenz first found the chaotic attractor in 1963,the theory of chaos has ob-tained the unprecedented development in many fields,such as secure communications,neural network,nonlinear circuits,mathematics and so on.The investigation of hyper-chaos is a new subject based on chaotic system.Compared with chaotic system,hyper-chaotic system has more complex dynamical behavior with at least two positive Lyapunov exponents.System’s randomness and indeterminacy has enhanced much.Hence hyper-chaos has much more extensive value in engineering application.This paper reports a new five-dimensional hyperchaotic system with three positive Lyapunov exponents,which is generated by adding a linear controller to the second e-quation of a 4D system that is obtained by coupling of a 1D linear system and a 3D mod-ified generalized Lorenz system.This 5D hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors.Of particular interest are the ob-servations that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under a unique equilibrium,three or infinite equilibria,and there are several types of coexisting attractors of this new 5D hyperchaotic system.Numerical analysis of phase trajectories,Lyapunov exponents,bifurcation,Poincare projections and power spectrum verifies the existence of the hyperchaotic and chaotic attractors.More-over,stability of hyperbolic or nonhyperbolic equilibrium and two complete mathematical characterization for 5D Hopf bifurcation are rigorously studied.The specific works of this paper are as follows:In the first chapter,we introduce the background of this paper and the current research at homeland and abroad,including the development phase of chaos and hyper-chaos theory.Then,we list the method and the theory which will be used in this paper.Meanwhile,the research in hyperchaotic systems are summarized.In the second chapter,based on 3D modified generalized Lorenz systema new 5D autonomous hyperchaotic system with three positive Lyapunov exponents is introduced.Meanwhile,Lyapunov exponents,hyperchaotic behaviors are further numerical analyzed.Moreover,the 5D hyperchaotic system is implemented visa electronic circuits,showing very good agreement with the simulation results.In the third chapter,the dynamical behaviors of this 5D hyperchaotic system such as the stability of hyperbolic or nonhyperbolic equilibrium are theoretical analyzed.More-over,we study the Hopf bifurcation by employing the high-dimensional Hopf bifurcation theory and applying symbolic inference.The expression and the direction of the bifur-cating periodic solution is investigated,too.In the forth chapter,we discuss the global dynamic behavior of the 5D hyperchaotic system.At first,according to the techniques of Lyapunov exponents spectrum,bifurca-tion diagram,Poincare map,we investigate the dynamic behaviors of this hyperchaotic system.When the system choose appropriate parameters,this system can exhibit hyper-chaotic,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 sys-tem,such as hyperchaotic and chaotic attractors,two chaotic attractors and so on.