Singular Orbits And Bifurcations For Several High Dimensional Chaotic Systems

Chaos, as a ubiquitous nonlinear phenomenon with complex dynamics in na-ture, has attracted the attention of scientists and engineers from various fields. Lorenz system——the first chaotic mathematical and physical model——and the relevant Lorenz-like system family has greatly promoted the development of chaos science. Compared with low-dimensional chaotic systems, high-dimensional chaotic systems and their attractors show more complex dynamic behaviors and greater po-tentials in applications, and become an important research branch among nonlinear science recently.Based on the status quo of the research for Lorenz-type system family, this dis-sertation not only deeply digs out the rich dynamics of some known existing chaotic and hyperchaotic systems, but also proposes another two novel hyperchaotic sys-tems (one is four-dimensional, and the other is five-dimensional), and thoroughly explores their dynamic behaviors. Precisely speaking, by employing the dynamic theories and methods, such as center manifold theory, normal form theory, bifurca-tion theory, project method, Poincare compactification, Lyapunov function method, numerical simulation and so on, this paper discusses both the local dynamics includ-ing the distribution, the stability and the bifurcation of equilibria, and the global dynamics consisting of the existence of homoclinic and heteroclinic orbits, singular-ly degenerate heteroclinic cycle, hyperchaotic attractor coexisting the non-isolated equilibria, hyperchaotic attractor bifurcated from singularly degenerate heteroclinic cycle and so on.The main research work in this disertation is organized as follows.In Chapter 1, some background and previous work on our research topic is introduced.In Chapter 2, chaos theory and the corresponding research techniques are sim-ply summarized.In Chapter 3, some dynamical behaviors not considered for a given three- dimensional Lorenz-like system has been deeply investigated. By employing some useful tools, i.e. the parameter-dependent bifurcation theory, normal form theorem, Lyapunov function, project method, Poincare compactification and so on, a more complete description of its dynamics is presented in the parameter space, not only from the "local" and "global", but also from the "finiteness" and "infinity". Further-more, numerical simulation shows the correctness of the corresponding theoretical results.In Chapter 4, a new four-dimensional autonomous hyperchaotic unified Lorenz-type system is introduced, which contains some famous existing systems as special cases. Utilizing the Routh-Hurwitz criterion, the parameter-dependent center man-ifold theory and bifurcation theory, its local dynamics is discussed, including the stability, fold bifurcation, pitchfork bifurcation and Hopf bifurcation of its equilib-rium. Combining the Lyapunov function and the definitions of a-limit set, w-limit set, it is rigorously proved that there exist two and only two heteroclinic orbits but no homoclinic orbits under some given suitable choice of its parameters. In addi-tion, the case of nonexistence of heteroclinic orbit is presented. In particular, it is numerically found that the hyperchaotic attractors are not created as its singularly degenerate heteroclinic cycles collapse.In Chapter 5, the undetected and rich dynamics of a complex Lorenz system is excavated, such as all the circle of equilibria being non-hyperbolic, the nonexistence of hyperchaotic attractor near the infinitely many singularly degenerate heteroclinic cycles, the existence of an infinite set of circle-type heteroclinic orbits to its origin and circle of equilibria, and so on.In Chapter 6, a novel five-dimensional autonomous hyperchaotic system is con-structed based on the Shimizu-Morioka system. Combining the theoretical analysis and numerical technique, it is found that the system has the interesting and distinct properties as follows:1. the existence of both ellipse-parabola-type and hyperbola-parabola-type of equi-libria;2. the hyperchaotic attractor coexisting the non-isolated equilibria;3. the existence of hyperchaotic attractor bifurcated from singularly degenerate heteroclinic cycle;4. the existence of an infinite set of both ellipse-parabola-type and hyperbola-parabola-type heteroclinic orbits with the cantor-like set of parameters.