Research On Complex Dynamics Of High-dimensional Segmented Disc Dynamo Type Hyperchaotic Systems

As a complex dynamic phenomenon in nonlinear dynamical system,chaos exists in nature widely.Chaos theory and its applied research has become one of the most attractive subjects in the nonlinear science.Compared with chaos,hyperchaos has stronger randomness and unpredictability,thus it has more complex dynamical properties.Hidden hyperchaotic attractors are important in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in a structure like a bridge or an airplane wing.Therefore,in the practical application,hyperchaotic system has a broad prospect for development.Segmented disc dynamo is a 3D quadratic Stokes flow about electromagnetic field,in which the current associated with the radial diffusion of the magnetic field could be included in a simple way.The magnetic field exists inside celestial bodies and in interstellar space between them widely so the study of magnetic field has the vital significance in astrophysics.With the deepening of the research,it has been found that 3D chaotic system has certain limitations in describing electromagnetic field,so we turn to high dimensional hyperchaotic system.Based on the SDD system,this paper put forward high dimensional SDD type hyperchaotic systems,and analyze their dynamical properties deeply.Using center manifold theory,normal form theory,bifurcation theory and Lyapunov function method,this paper investigates the stability and bifurcation of the equilibrium,the ultimate bound of hyperchaotic system,the existence of hyperchaotic attractor,chaotic attractor,periodic attractor,and the coexisting phenomena of attractors.The main research works are as follows:In Chapter 1,the research background and significance of this paper are presented.The research history and achievement of chaos,hyperchaos theory are introduced.The SDD system,hidden attractor and the ultimate bound are enumerated.In Chapter 2,a 4D segmented disc dynamo type hyperchaotic system is proposed based on segmented disc dynamo.Lyapunov exponents spectrum and bifurcation diagram characterize the existence of hyperchaos in the system quantitatively.When the system has a line of equilibria or a stable equilibrium,it possesses coexistence of hidden hyperchaotic and chaoticattractors.With the help of the parameter-dependent center manifold theory and bifurcation theory,the local dynamics,including the stability,Hopf bifurcation and pitchfork bifurcation of equilibrium of this system are studied.The ellipsoidal ultimate bound sets of this system are estimated by combining the Lyapunov function method and appropriate optimization method,so the global dynamical properties of the system are obtained.Some numerical investigations are also exploited to demonstrate and visualize the corresponding theoretical results.In Chapter 3,a 5D segmented disc dynamo type hyperchaotic system with three positive Lyapunov exponents is proposed.Interestingly,when the system has a line of equilibria,a stable equilibrium or no equilibrium,it possesses coexistence of hidden hyperchaotic and chaotic attractors.When the system has an unstable equilibrium,it possesses coexistence of self-excited hyperchaotic and chaotic attractors.With the help of geometric theory of differential equation,the center manifold theory and bifurcation theory,Hopf bifurcation and pitchfork bifurcation of this system are studied,and the conditions and the directions of them are obtained.The ultimate bound set of this system are obtained by combining the Lyapunov function method and appropriate optimization method,and the location for hyperchaotic attractor of this system is estimated effectively.Some numerical investigations are also exploited to demonstrate the corresponding theoretical results.Moreover,the system is also implemented with electronic components.