Research On Complex Dynamics Of New Four-dimensional Hyperchaotic System Based On Lorenz-like System

Since 1979 R(?)ssler proposed the first four-dimensional nonlinear hyperchaotic system,hyperchaos has attracted many scientists to study the upsurge.Hyperchaotic systems are widely concerned because they have more complex dynamical behaviors than chaotic systems,that is,they have at least two positive Lyapunov exponents,which means that hyperchaotic attractors expand in two or more directions at the same time.Because of the complex dynamic characteristics of hyperchaotic attractors,hyperchaotic systems have greater application value than chaotic systems.For example,in many engineering fields,especially in secure communication,hyperchaos has the ability to improve the security of communication systems.In chaos theory,equilibrium point plays an important role in understanding the complex dynamic behavior of the system.Therefore,in this paper,a new four-dimensional nonlinear autonomous system with only two simple quadratic terms is proposed.There are hyperchaotic attractors in all four cases of the system with no equilibrium point,only one equilibrium point,two equilibrium points and three equilibrium points.At the same time,there are many kinds of complex dynamical phenomena,such as coexistence of attractors.The research contents of this paper are as follows:In chapter 1,the research background and significance of this paper are introduced.Firstly,the background and significance of chaos and hyperchaos are understood by briefly describing the development of chaos.Secondly,the related theory and analysis method of chaos are given,which provides a theoretical basis for further study of hyperchaos.Finally,several typical hyperchaotic systems and the research on hyperchaos in recent years are introduced.In chapter 2,based on the Lorenz-like system,a new four-dimensional hyperchaotic system with only two simple quadratic terms is proposed by using the linear feedback control technique.Furthermore,there exist hyperchaotic attractor with two positive Lyapunov exponents in all four cases of the system with no equilibrium point,only one equilibrium point,two equilibrium points and three equilibrium points.The existence of this phenomenon is verified by numerical simulation.In chapter 3,the local dynamics of the new four-dimensional hyperchaotic system is studied.First,the stability of the hyperbolic equilibrium point of the new four-dimensional hyperchaotic system is analyzed by using the Routh-Hurwitz criterion.Secondly,the existence of Hopf bifurcation at the equilibrium point of the system is analyzed by using the high dimensional Hopf bifurcation theory,and the stability and approximate expression of the bifurcation periodic solution are obtained.Finally,the fork bifurcation at the non-hyperbolic equilibrium point is studied by using the center manifold theorem and the normal form theory,and the dynamic behavior near the non-hyperbolic equilibrium point is obtained.In chapter 4,the global dynamics of the new four-dimensional hyperchaotic system is studied.By maens of phase diagram,Poincar(?) map,Lyapunov exponent spectrum and bifurcation diagram,it is verified that the new four-dimensional hyperchaotic system has a hyperchaotic,chaotic,quasi-periodic and periodic attractor in all four cases where there is no equilibrium point,only one equilibrium point,two equilibrium points and three equilibrium points.At the same time,it is found that under the same set of parameters,there are many kinds of complex dynamic phenomena such as the coexistence of attractors when different initial values are selected.