Complex Dynamic Of A New Three-dimensional Chaotic System With Infinite Isolated Singularities

Lorenz, a famous meteorologist in the United States, discovered the chaotic attrac-tor for the first time during the research on Meteorological convection problem. Lorenzsystem, as the chaotic research precedent, has occupied an important position in non-linear science investigative history. Chaos not only widely exists in nature, but also aspecific phenomenon in nonlinear science, namely subjects to a certain equations, but tosome extent, shows an analogy randomicity. Chaos, as an important branch of nonlinearscience, research on its complex dynamics has attracted deeply mathematics and relat-ed subjects. What’s more, chaos has also some great application potential in nonlinearcircuits, communications confidential, etc.In the paper, a new three-dimensional chaotic system with infinite isolated singu-larities is proposed and mainly study its complex dynamics and synchronization throughusing the center Manifold theorem and the Normal Forms theorem. The main researchcontents are as follows:In Chapter 1, The Introduction is shown. In this part the research backgroundsand significance are stated, and the development history and research status of chaos arealso briefly introduced. In the end, some preliminary knowledge and research tools areillustrated.In Chapter 2, Three-dimensional chaotic system are classified with taking the quan-tities of singularities as the main line and some classical three-dimensional chaotic systemare introduced. Mainly includes chaotic systems with finite isolated singularities, chaoticsystem with a line of equilibrium or a curve of equilibrium and chaotic system with noequilibrium. What’s more, a briefly review and analysis are made about their basic dy-namics. Going along the train of this analysis, a new three-dimensional chaotic systemwith infinite isolated singularities will be introduced. Meanwhile, the established pro-cess and the research value are stated. In addition, some basic dynamics are analyzed,including fractal dimension, dissipation, sensitively dependence on initial conditions andthe existence of equilibrium.In Chapter 3, Complex dynamic of a new three-dimensional chaotic system is pri-marily investigated. The stability of hyperbolic equilibrium are studied, Hopf bifurcationare analyzed under two di?erent bifurcation parameters by using Hopf bifurcation theory,including the stability of Non-hyperbolic equilibrium and the limit cycle, the directionof bifurcation and the condition of the degenerated Hopf bifurcation. In addtion, globalcomplex dynamic of new system including parameter interval of chaotic attractor, periodattractor and numerical characteristics are also analyzed through using phase diagram,Lyapunov exponents and ISI etc. Meanwhile, the existence of chaotic attractor and periodattractor are verfied through the phase diagram.In Chapter 4, The synchronization of new system is investigated. Some synchroniza-tion method are introduced, including the driven-response synchronization method, theactive and passive Synchronous method and the variable control synchronization method.Meanwhile, Using these methods and adding the appropriate controller which makes thesynchronization become a reality between the responsive system and the driven system.