The Research On The Problems Of Bifurcations And Chaos In Several Classes Of High-dimensional Nonlinear Systems

In physics, chemistry, biology and engineering, the governing equations of motion for a number of problems can be described by high-dimensional nonlinear systems. Comparing with linear systems and low-dimensional nonlinear systems, the study on the dynamical behavior of high-dimensional nonlinear systems were more difficult, so there are more complexities and richer in the dynamical phenomena of high-dimensional nonlinear systems. High-dimensional nonlinear systems have received considerable attention and play an important part in this area, especially, the study of bifurcations and chaos of high-dimensional nonlinear systems.By using normal form method, global perturbation method, energy-phase method, Silnikov method and numerical methods, the present dissertation is devoted to the bifurcations and chaos of some high-dimensional nonlinear dynamical systems.Firstly, both normal form method and numerical approaches are employed to consider the stability and local bifurcation for the flexible beam system undergoing a large linear motion with combination parametric resonance and 3:1 internal resonance. We discuss the stability and local bifurcation for the flexible beam to a principal parametric excitation of either the first or the second mode or a combination parametric resonance of both modes. The critical bifurcation curves, bifurcation solutions and their stabilities are obtained. Some significant dynamical phenomena are obtained. Numerical simulations agree with the analytic prediction.Secondly,with the global perturbation method and the energy-phase method, we discuss the global bifurcations and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators. Using the global perturbation method, the existence of Silnikov-type single-pulse orbits and chaotic motions of Silnikov-type single-pulse of two non-linearly coupled parametrically excited van der Pol oscillators are studied in detail. And using the energy-phase method, the existence of Silnikov-type multi-pulse orbits of these systems are studied in detail. These results show that the chaotic motions of Silnikov-type multi-pulse can occur. Some new significant dynamical phenomena are obtained.Thirdly, Silnikov-type single-pulse orbit, Silnikov-type multi-pulse orbits and chaotic motions of two types of nonlinear suspended cables systems are studed in detail in this dissertation. The global perturbation method is applied to study the existence of Silnikov-type single-pulse orbits and chaotic motions of Silnikov-type single-pulse of two types of nonlinear suspended cables systems. The energy-phase method is utilized to analyze the existence of Silnikov-type multi-pulse orbits of these systems. These results show that the chaotic motions of Silnikov-type multi-pulse can occur.Lastly, for Rucklidge systems and a new 3-D quadratic autonomous system with a four-wing chaotic attractor, Hopf bifurcation and Silnikov chaos are discussed in detail with the first Lyapunov coefficient and Silnikov method. Using undetermined coefficient method, the existence of heteroclinic and homoclinic orbits of these systems are proved, and the Silnikov criterion guarantees that there exists the Smale horseshoe chaos motion.