Complex Dynamical Analyses Of A Hyperchaotic Faraday Disk Dynamo

Chaos exists widely in nature.Chaos theory builds a bridge between the two theoretical systems of certainty theory and probability theory.It is called the three great revolutions of physics in the 20 th century together with quantum mechanics and relativity theory.The study of chaos is one of the most important contents in nonlinear science.Compared with chaotic systems,hyperchaotic systems have at least two positive Lyapunov exponents,so they have more complex dynamic behaviors and greater application potential.In recent years,the theory and application of hyperchaos have attracted great interest of scientists,and it has become a very important research field in nonlinear science.In order to study the magnetic fields between celestial bodies,Hide and Moroz proposed a class of hyperchaotic Faraday disk dynamo,which describes the action of azimuthal eddy currents.At present,the existing results are all through numerical analysis to study its dynamic properties,drawing shows the existence of Hopf bifurcation.But it doesn't give a rigorous mathematical proof.In this paper,the stability,multistability,bifurcation and Jacobi stability of the equilibrium point of the system are studied by using central manifold,canonical form and Kosambi-Cartan-Chern(KCC)theory.The existence of Hopf bifurcation is not only proved mathematically strictly,but also many new properties of the system are discovered.These studies provide a new way to explain the mechanism of chaos and are of great value to the study of magnetic field flow in celestial bodies.The main content of this paper is as follows:The first chapter is Introduction,which briefly introduces the research background of this paper,the current research status at home and abroad,including the background of chaos and hyperchaos theory,and summarizes the used methods and theories.The second chapter studies the stability of the equilibrium points and multistability.Under different parameters,it is found that the system has complex dynamical behaviors,such as hyperchaos,chaos,as well as the coexistence of self-excited hyperchaotic attractor and chaotic attractor.In Chapter 3,the pitchfork bifurcation,Hopf bifurcation and zero-zero-Hopf bifurcation are studied by using the center manifold theory with parameters,Hopf bifurcation theory and average theory.The conditions for the existence of bifurcations are given,and numerical simulations are performed on the obtained results.In Chapter 4,the Jacobi stability of the equilibrium point and the periodic orbit are analyzed by KCC theory.At the same time,we use deviation vector to study the orbit behavior near the equilibrium point and the periodic orbit.Instability exponents have been studied to explain the onset of chaos.