Dynamic Analyses Of A New 4D And 5D Hyperchaotic Systems Based On Segmented Disc Dynamo

Hyperchaos and chaos are two complicated phenomena of nonlinear dynamical systems.Because of the randomness and unpredictability of chaotic behavior,chaos theory has been widely applied and developed in the fields of secure communication and nonlinear circuit.Segmented Disc Dynamo is a kind of autonomous dynamical system describing the principle of magnetic field formation and reversal between celestial bodies.Based on this system,a new four-dimensional hyperchaotic system and a new five-dimensional hyperchaotic system are proposed in this paper.From the point of view of local dynamics,the two new systems can have any given number of equilibria,and both have Hopf bifurcation phenomenon.Interestingly,under the condition of appropriate parameters,the new four-dimensional system will simultaneously produce Hopf bifurcation at two equilibrium points,and then generate two symmetric limit cycles.From the point of view of global dynamics,there are many kinds of coexisting attractors in both new systems.In particular,both new systems have megastability phenomenon,the coexistence of an infinite number of hidden attractors.The research contents of this paper are as follows:The first chapter is the introduction,which briefly describes the development of chaos dynamics theory,outlines the basic concepts of chaos,central manifold theory and Hopf bifurcation theory,and briefly explains the structure of this paper.In the second chapter,a new four-dimensional system and a new five-dimensional system were proposed based on Segmented Disc Dynamo.Among them,the new four-dimensional system has a hidden chaotic attractor under no equilibrium or line equilibria,a hyperchaotic attractors under two non-hyperbolic equilibria,a chaotic attractors under six unstable equilibria or infinite number of unstable isolated equilibria.The new five-dimensional system has a hyperchaotic attractor in the case of three unstable hyperbolic equilibria.In the third chapter,the dynamic behavior of the new four-dimensional hyperchaotic system is analyzed.It is proved that the system can have any given number of equilibria,including no equilibria,any finite equilibria,infinite isolated equilibria and line equilibria.Furthermore,the stability of the equilibria in some cases is analyzed.The Hopf bifurcation theory is used to prove that the new four-dimensional system can produce two symmetric limit cycles at two equilibrium points at the same time under the appropriate parameter conditions,and the existence of Hopf bifurcation is verified by numerical simulation.The new fourdimensional system also have some kinds of multistability phenomenon,including the coexistence of two hyperchaotic attractors and the coexistence of infinitely many chaotic attractors.What is particularly interesting is the phenomenon of megastability in the new fourdimensional hyperchaotic system.In chapter 4,the dynamic behavior of the new 5-dimensional hyperchaotic system is analyzed from the same point of view as in chapter 3.The new five-dimensional system can also have any given number of equilibria,and Hopf bifurcation also exists under appropriate parameter conditions.In addition,the new five-dimensional system also have some kinds of mulistability phenomenon,including the coexistence of a chaotic attractor and a periodic attractor and the coexistence of a hidden quasi periodic attractor and a hidden periodic attractor,and the phenomenon of megastability.