Research On The Complex Dynamics Of A New Five-Dimensional Hyperchaotic System With Infinite Isolated Attractors

Nonlinear dynamic systems have very rich dynamic properties,and chaotic attractors occupies an important position in dynamic systems.Chaotic attractors have become the frontier research of dynamic systems due to their complex structure and potential applications in many fields such as engineering and security communications.Compared with chaotic phenomena,hyperchaotic systems present more complex dynamics.For a long time,scholars have been studying finite-dimensional autonomous ordinary differential dynamic systems that can generate only a finite number of chaotic or hyperchaotic attractors.In two-dimensional planar polynomial systems,especially those system which with finite isolated equilibrium points.It is very difficult to study how many isolated attractors and limit cycles there are.Therefore,in three-dimensional and higher-dimensional dynamic systems,it is more difficult to study the inner relationship of number of isolated singularities and chaos or hyperchaos.This is an interesting question that is rarely studied by scholars.Based on the coupling technology and the design of linear feedback controllers,this paper proposes a new five-dimensional hyperchaotic system with infinitely number of isolated equilibrium points or no equilibrium points.By studying its dynamic characteristics,it is pointed out that this new five-dimensional system has infinitely number of coexisting isolated hyperchaotic attractors or infinite number of coexisting isolated hidden hyperchaotic attractors.At the same time,using the Routh-Hurwitz criterion,the canonical form theory,the central manifold theorem,and the geometric theory of differential equations,the new system is analyzed: the stability of hyperbolic and nonhyperbolic equilibrium points and the Hopf bifurcation problems at an infinite number of equilibrium points,and further discuss the global dynamics characteristics of the new system.The specific research contents of this paper are as follows.Chapter 1 is the introduction,which mainly describes the development process and research status of chaos science,and summarizes the definition of chaos and the basic theories and methods of chaos research.At the same time,it briefly introduces several types of typical four-dimensional hyperchaotic systems,as well as the research status of five-dimensional hyperchaotic systems.Chapter 2 proposes a new type of five-dimensional hyperchaotic system with infinitely many isolated equilibrium points or no equilibrium points.No matter in the case of infinitely many isolated equilibrium points or no equilibrium points,the new system can produce three positive Lyapunov exponential hyperchaotic attractor.And through numerical simulation,it is found that in the case of infinitely many isolated equilibrium points,there are infinitely many isolated coexisting hyperchaotic attractors.And in the case of no equilibrium points,there are infinitely many isolated coexisting hidden hyperchaotic attractors.Chapter 3 uses the related theories and methods of studying chaos to analyze the local dynamic properties of the new five-dimensional hyperchaotic system,and discusses the dissipation of the system,the existence of equilibrium points,and the stability of hyperbolic and non-hyperbolic equilibrium points.At the same time,using the highdimensional Hopf bifurcation theory and strict symbolic reasoning,the existence and stability of the Hopf bifurcation and the corresponding bifurcation direction of the new system are discussed.The approximate expressions and approximate periodic solutions of the period and characteristic exponent of the Hopf bifurcation are further given.Chapter 4 uses Lyapunov exponential spectrum,bifurcation diagram,phase diagram,and Poincarémapping to explore the global dynamics of the new five-dimensional system and study its complex dynamics.Under certain parameters,the new system has hyperchaos,Chaos,cycle and other complex dynamic phenomena.At the same time,numerical simulations are used to verify that the attractors of the new system are periodic in the direction,and six types of attractors coexist under different parameters in the new system.