Research On Complex Dynamics Of Four-dimensional Hyperchaotic Systems Based On Lorenz-type Systems

As a complex dynamic phenomenon in nonlinear dynamical system, chaos exists innature widely. Due to its complex dynamical behavior, chaos has attracted the greatinterest of scholars from various fields and areas. Chaos theory and its applied researchhas become one of the most attractive subjects in the nonlinear science. As the firstmathematical and physical model of chaos, Lorenz system has the milestone significancein the history and planted the seed for the new theory of chaos science. The researchof Lorenz-type systems runs through the whole developing process of chaos science. Hy-perchaos, as a dynamic behavior, is far more complex and has a greater potential inapplications than chaos. Recently, the hyperchaos theory and its applied research hasbecome one of the most important research fields in the nonlinear science.Based on Lorenz-type systems, this paper proposes several new four-dimensional hy-perchaotic systems, and investigates the dynamic behaviors of these systems in depth.Concretely, using the dynamic theories and methods, such as center manifold theory,normal form theory and Lyapunov function method, this paper investigates the local dy-namics including stability and bifurcation of equilibrium, and global dynamics includingthe ultimate bound of hyperchaotic system, the existence of homoclinic and heteroclinicorbits, hyperchaotic attractor, chaotic attractor, quasi-periodic attractor, periodic attrac-tor, singular degenerate heteroclinic cycle, and the coexisting phenomena of attractors.The main research works are as follows:In Chapter1, the research background and significance of this paper are presented.The research history, achievement and basic knowledge of chaos theory are introduced.The classical Lorenz system, Lorenz-type hyperchaotic systems and some other typicalhyperchaotic systems are enumerated.In Chapter2, a generalized four-dimensional Lorenz-Stenflo chaotic system is pro-posed. With the help of the parameter-dependent center manifold theory and Hopfbifurcation theory, the local dynamics, including the stability and bifurcation of equilib-rium, of this system are studied. The ultimate bound sets of this system are obtainedby combining the Lyapunov function method and appropriate optimization method, andthe location for arbitrary attractor of this system is estimated efectively.In Chapter3, a four-dimensional autonomous Lorenz-type hyperchaotic system isinvestigated. Using the parameter-dependent center manifold theory and Hopf bifurca-tion theory, the local dynamics of this hyperchaotic system, such as the stability andbifurcation of equilibrium, are investigated. Under a certain parameter condition, the ex- istence of homoclinic and heteroclinic orbits of this hyperchaotic system is further studiedrigorously, and the fact that this hyperchaotic system has two symmetrical heteroclinicorbits but no homoclinic orbit is proved.In Chapter4, a new four-dimensional Lorenz-type hyperchaotic system with no equi-librium is proposed. Utilizing the techniques of bifurcation diagram, Lyapunov exponentsspectrum and Poincar′e map, the dynamic behaviors of this hyperchaotic system are in-vestigated thoroughly. Choosing proper parameters, this new system can exhibit hyper-chaotic, chaotic, quasi-periodic and periodic dynamics. In particular, several alternativecoexisting attractors of this new hyperchaotic system can be observed, such as hyper-chaotic and periodic attractors, quasi-periodic and periodic attractors, diferent periodicattractors, et al. Moreover, the technique of Poincar′e compactification is used to inves-tigate the dynamics at infinity of this hyperchaotic system with no equilibrium.In Chapter5, a new four-dimensional Lorenz-type hyperchaotic system with a curveof equilibria is proposed. Choosing proper parameters, this new system can display hyper-chaotic, chaotic, quasi-periodic, periodic dynamics and singular degenerate heterocliniccycles. In particular, with the exception of curve of equilibria, there are several otheralternative coexisting attractors in the phase space of this new hyperchaotic system, in-cluding chaotic and quasi-periodic attractors, chaotic attractor and singular degenerateheteroclinic cycle, periodic attractor and singular degenerate heteroclinic cycle, diferentperiodic attractors, et al. The possible existing ranges of singular degenerate heterocliniccycles of this hyperchaotic system is obtained under a certain parameter condition, andthe locations of these equilibria, which are connected by these singular degenerate het-eroclinic cycles, are obtained. With the help of numerical simulation, lots of singulardegenerate heteroclinic cycles are found.