Research On Complex Dynamics Of Three-dimensional Nonlinear Autonomous Systems

As is well-known,there exist complex dynamical behaviors in nonlinear dynamical systems,where the research of dynamics focusing on bifurcation and chaos has become a significant field in nonlinear science.As the first mathematical model of chaos,Lorenz system was regarded as the milestone in the history of chaos.After that,chaos theory and bifurcation theory in nonlinear systems were improved continually and an increasing number of new research results came forth.Also,chaos theory and bifurcation theory had broad applications on control theory,information science,bioscience,engineering and so on.Based on Lorenz system,this paper first proposes a three-dimensional quadratic system and investigates properties of complex dynamics including bifurcation and chaos.Then,this paper studies a generalized Langford system describing the characteristics of turbulence,and analyzes periodic orbits,heteroclinic cycles,and other complex dynamics.Last,this paper finds a nonlinear analytical system which has three different types of infinitely many chaotic attractors.The primary research work is as follows.Chapter 1 presents the research background and significance of this paper.The history of development and achievements of chaos theory are introduced briefly.Basic methods of studying nonlinear systems are summarized.Some typed three-dimensional nonlinear autonomous systems are enumerated.In Chapter 2,a modified Lorenz-Like system is proposed,and the inequivalence with classical Lorenz system is proved.The pitchfork bifurcation and Hopf bifurcation are investigated,and also the approximate expression and the stability of the bifurcating periodic orbit are given.Under certain conditions,an invariant algebraic surface is obtained,and the dynamics on the surface are further analyzed.By numerical simulations,three kinds of two coexisting attractors are obtained,including one-scroll chaotic attractor and periodic attractor,two-scrolls chaotic attractor and periodic attractor,two periodic attractors.In Chapter 3,a generalized Langford system is studied.Using center manifold theory,the stabilities of the origin are investigated completely,including hyperbolic and nonhyperbolic case.The Hopf bifurcation,as well as the approximate expression and the stability of bifurcating periodic orbit,are analyzed.Under some conditions,another kind of the periodic orbit is obtained,and also the accurate expression and the stability of such periodic orbit are given.Combining with the qualitative method,this chapter proves the existence of two heteroclinic cycles consisting of three heteroclinic orbits,and the coexistence of such two heteroclinic cycles and the accurate periodic orbit is also obtained under certain conditions.By numerical simulations,a kind of four coexisting attractors and two kinds of two coexisting attractors are observed,including four coexisting periodic attractors,two coexisting periodic attractors(one with accurate expression),coexisting of quasi-periodic attractors(invariant torus)and periodic attractor.In Chapter 4,a three-dimensional autonomous system with no equilibria or infinitely many isolated equilibria is proposed.Interestingly,there exist three different types of infinitely many chaotic attractors in such a system.The stabilities of the hyperbolic and nonhyperbolic equilibria are analyzed completely.Furthermore,the system with no equilibria has infinitely many coexisting hidden chaotic attractors and infinitely many coexisting hidden periodic attractors.The system with infinitely many nonhyperbolic equilibria has infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors.The system with infinitely many hyperbolic and nonhyperbolic equilibria has infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors.