Research On Dynamics Of A New 4D Hyperchaotic System With Infinite Isolated Equilibria Or No Equilibria

As a complex dynamic phenomenon in nonlinear dynamical system,chaos exists in nature widely.The research and applications of chaos has become one of the most attractive subjects in the nonlinear science.A chaos system with two or more positive Lyapunov exponents is known as hyperchaos system.Hyperchaos,as a dynamic behavior,is far more complex and has a greater potential in applications than chaos,which makes it have more difficulty in studying and very few discussions.So that to attract the more study in mathematics and related fields.Recently,the research of hyperchaos system is mainly centered on four dimensional systems with finite isolated equilibria.There are very few discussions in four dimensional systems with infinite non-isolated equilibria(a line of equilibria or a curve of equilibria)or no equilibria,especially for the four dimensional systems of the two types of equilibria of infinite isolated equilibria or no equilibria.Based on a three dimensional chaotic system with infinite isolated equilibria,by introducing a linear feedback controller to the chaotic system,this paper discusses a new four dimensional hyperchaotic system with infinite isolated equilibria or no equilib-ria.Further analysis the local dynamics of the new system,the determinant conditions of stability of non-hyperbolic equilibria is obtained by using the center manifold theory.Meanwhile,this new system,which can display hyperchaotic,chaotic and periodic attrac-tors is investigated through numerical simulation.Of particular interest is that there are five types alternative coexisting attractors of this new hyperchaotic system with infinite isolated equilibria or no equilibria.The mains works are as follows:In Chapter 1,the research background and significance of this paper are introduced.The origin,historical development and research status of chaos theory are presented.Then we list some basis knowledge and methods about chaos theory.In Chapter 2,based on the number of equilibrium,some typical four dimension-al hyperchaotic systems,including hyperchaotic systems with finite isolated equilibria,hyperchaotic system with a line of equilibria or a curve of equilibria and hyperchaotic system with no equilibria,are introduced respectively.And the basic dynamic properties of these systems are briefly analyzed.A new four dimensional hyperchaotic autonomous system is proposed,under a certain condition,two types of equilibria of infinite isolated equilibria or no equilibria coexist in a same system.At last,with the help of numerical simulation,the new system can exhibit hyperchaotic attractor and hidden hyperchaotic attractor,et al.In Chapter 3,we main discuss some basic properties,such as dissipativity,sensitive dependence with the initial value and the existence of equilibrium.Meanwhile,other local dynamics,including the stability of hyperbolic equilibria and non-hyperbolic equilibria and the existence of Hopf bifurcation,of this system are obtained.In Chapter 4,according to the techniques of bifurcation diagram,Lyapunov expo-nents spectrum and Poincare map,the global dynamic behaviors of this hyperchaotic system are investigated thoroughly.Choosing proper parameters,this new system can display hyperchaotic,chaotic,periodic,hidden hyperchaotic,hidden chaotic and hidden periodic dynamics.In particular,with the exception of infinite isolated equilibria or no equilibria,there are several other alternative coexisting attractors of this new hyperchaot-ic system,such as hyperchaotic and periodic attractors,chaotic and periodic attractors,different periodic attractors,hidden hyperchaotic and hidden periodic attractors,hidden chaotic and hidden periodic attractors,et al.