Dynamical Analysis Of A Class Of Three-dimensional Chaotic System

Chaos, existing widely in the nature, is a kind of complicated phenomena in nonlineardynamic systems, and a motion that seems to be random occurred in the deterministicsystem. Since M.O.Lorenz proposed the first chaotic model, chaos has attracted more andmore attention, and new achievements emerged constantly in study of chaos theory andapplication. These achievements have been applied in engineering, information science,biological science and other fields, it also has broad application prospects in sociology.This paper proposes a three-dimensional autonomous chaotic system with five pa-rameters and three quadratic terms, and investigates the dynamic behaviors of the system.After giving the proof of inequality of the system and the unified-Lorenz system underthe condition that the new system has three equilibrium points, by using the parameter-dependent center manifold theory and bifurcation theory, this paper investigates the localdynamics including stability, pitchfork bifurcation and Hopf bifurcation of equilibriumsand the global dynamics including the singular degenerate heterolinic orbits. At the sametime, by using the Poincar′e compactification, the paper also investigates the dynamicsbehavior in the infinity. The main research works are as follows:In the first chapter, the research background and the significance of this paper aremainly introduced. The development history, basic concepts of the chaos, the local bi-furcation theory and determination method of chaos are introduced and several kinds ofclassical chaotic system are enumerated.In the second chapter, a class of chaotic system are proposed. By using the Lyapunovspectrum and Bifurcation diagram and some other tools, it is verified that the systemcan produce chaos in the appropriate parameters, After giving the proof that the newsystem and the unified-Lorenz system system are smoothly nonequivalent under the con-dition that the new system has three equilibrium points, the stability of the hyperbolicequilibrium are studied and obtain the judging condition of stability.In the third chapter, the local dynamic properties of nonhyperbolic equilibrium pointof the system are studied. Before discussing the stability of equilibrium point with twozero characteristics, with the help of the parameter-dependent center manifold theory,bifurcation theory and the normal form theory, the Pitchfork bifurcation and Hopf bi-furcation of the equilibrium point are analysed.In the forth chapter, the global dynamics of system are analysed. By using thePoincar′e compactification method, the infinity dynamic behaviors of the system are in-vestigated. Through numerical simulation, multiple singular degenerate heteroclinic or-bits are found in the specific parameters. In particular, the system has two straight linesof equilibrium points in the proper parameters. When disturbs specific parameters, thesystem emerges a chaotic attractor.