Research On Complex Dynamics Of A Five-dimensional Hyperchaotic System Based On Jerk Equation

Since Sprott proposed Jerk system at the end of the 20 th century,Jerk system has aroused a research boom in the field of non-linear science,because its dynamic characteristics can be used to describe complex variable acceleration motions,and it has the advantage of easy circuit realization.Because of these characteristics,Jerk system is widely used in the field of physics and engineering,especially in the field of quenching force dynamics and mechanical oscillator design.At present,researches on Jerk system are mainly focused on the three and four dimensions.There are few discussions about the high-dimensional Hyperjerk system,and the high-dimensional Hyperjerk system is also of great research value in engineering applications.Therefore,based on the three-dimensional Jerk chaotic system,a class of five-dimensional hyperchaotic Hyperjerk systems with one,two and three equilibria is obtained by combining feedback control and coupling techniques.In these three cases,the system can generate hyperchaotic attractors that expand in three directions.The specific research contents of this paper are as follows:The first chapter mainly focuses on the research background and development status of Jerk system,combined with the development process of chaos dynamical system,and points out the significance and value of this research.This paper summarizes the chaos theory and analysis methods mainly used in this paper,and introduces several typical Jerk systems and Hyperjerk systems.In the second chapter,the linear feedback control technique is used to obtain the 4D hyperchaos Hyperjerk system based on the 3D Jerk system,and then on this basis,using the coupling technology,a 5D hyperchaotic Hyperjerk system is proposed.Further analysis of the dynamic evolution of the system under the independent variation of parameters(6,(8,(9,7)shows that the system can not only produce hyperchaos,chaos,quasi-period and periodic attractor,but also have the phenomenon of period-doubling bifurcation.In chapter 3,starting from the local dynamics of the system,the existence of the equilibrium point of the system is analysed firstly,and then the Routh-Hurwitz criterion is used to investigate the stability of hyperbolic equilibrium in Hyperjerk system.Then,the existence of Hopf bifurcation at the hyperbolic equilibrium point and the stability of bifurcation periodic solution are obtained by using the high-dimensional Hopf bifurcation theory.Finally,the existence of fork bifurcation at non-hyperbolic equilibrium point is analysed by combining fork bifurcation theory and central manifold theorem.In chapter four,the dynamics of the system is analysed from a global perspective.Firstly,the dynamics of the hyperchaotic Hyperjerk system with only one,two and three equilibria points are discussed.It is found that there are hyperchaotic attractors,quasiperiodic attractors,chaotic attractors and periodic attractors in the five dimensional hyperchaotic Hyperjerk system under these three conditions,and the phenomenon of double cycle bifurcation appears when the parameter (6 is changed separately.Further,the complex dynamic phenomena of the coexistence of two attractors with different types and two attractors with the same type are found.