Studies On Complicated Dynamics Of Six-Dimensional Nonlinear Systems

Since the 1960s, the basic research of nonlinear phenomenon was successfully developed from different branches in nonlinear sciences. Henceforth, it is regarded as"The Third Revolution"in natural sciences in the 20th century. The theories of nonlinear science are established from the multidisciplinary subjects in engineering, physics and mathematics. There frequently exist the chaotic motions in nonlinear dynamical systems. In reality, it is possible to find the steady state response, transient process and instable state of chaotic motions in practical problems.In engineering problems, the mathematical models and its dynamical equations can be governed by the high-dimensional nonlinear systems. Contrary to the low-dimensional nonlinear systems, the theory, geometric description and numerical simulation of high-dimensional ones are much more sophisticated. Heretofore, the analytical methods are often unavailable for studying the global bifurcation and chaotic dynamics of high-dimensional nonlinear systems. The development of theories and the provision of systematic applications to many engineering problems are very appealing and challenging. Hence, it is important to provide an all-embracing understanding of high-dimensional nonlinear systems, including the global bifurcation and the chaotic dynamics. It is a research forefront in nonlinear science as well.The Melnikov method and the energy-phase method are mainly used to probe the global bifurcation and chaotic dynamics of four-dimensional nonlinear systems. There have been numerous studies on the four-dimensional nonlinear system. In this dissertation, the global perturbation and energy-phase methods are improved to study the global bifurcation and chaotic dynamics of six-dimensional nonlinear mechanical systems. Numerical simulation is also given to verify the analytical solutions. The main research scopes of this dissertation are categorized as follows.1. The chaotic dynamics and jumping orbits of the six-dimensional nonlinear system are investigated for the first time, which represents the averaged equation of an axially moving viscoelastic belt in the case of 1:2:3 internal resonance.To simplify the six-dimensional averaged equation to a simpler normal form with the same topological equivalence, the theory of normal form and the method of the inner product are adopted. The global perturbation method is employed to analyze the existence of the homoclinic bifurcation and the single-pulse Shilnikov orbits for the six-dimensional nonlinear system. Using the energy-phase method, it is found that the system contains the multi-pulses jumping. The takeoff point is a fixed point in the invariant manifold and the landing point lies in the domain of the attraction of the fixed point. From the numerical results, the chaotic motions and jumping phenomena of orbits in the nonlinear system are demonstrated. Besides, it is found that the excitations f 1 and f 2 have significant effects on the nonlinear dynamical behavior of the belt.2. The dynamical characteristics of the six-dimensional nonlinear system are revealed for the first time, which is the nonlinear averaged equation of the composite laminated plate in the case of 1:2:3 internal resonances.First, the theory of normal form is used to simplify the six-dimensional averaged equation to a simpler normal form. Then, the global perturbation method is also employed to study the homorclinic bifurcation and the single-pulse Shilnikov orbits for the nonlinear system with small perturbation. in addition, The energy-phase method is extended to study the heteroclinic bifurcation and multi-pulse Shilnikov type orbits of the nonlinear system. From the numerical results, the chaotic motions and jumping phenomena of orbits for the composite laminated thin plate are discovered under certain conditions. It is also shown that the excitation f 3 performs significant effect on the nonlinear dynamical behaviors of the composite laminated thin plate. The motions of the composite laminated thin plate are changed to the particular sequence by increasing the forcing excitation f 3 as: the chaotic motion→the periodic motion→the chaotic motion→the periodic motion.3. By combining the theory of normal form and the energy-phase method, the multi-pulse and dynamic characteristics of the six-dimensional nonlinear system are studied for the first time. This high-dimensional system is used to describe the laminated composite piezoelectric rectangular plate in the case of 1:2:4 internal resonances.The theory of normal form is used to simplify the six-dimensional averaged equation to a simpler normal form with the same topological equivalence. The energy-phase method is extended to study the homorclinic bifurcation and the multi-pulse Shilnikov orbits for the nonlinear system with small perturbation. From the numerical results, it is illustrated that there exist the chaotic motions and jumping phenomena of orbits in the laminated composite piezoelectric rectangular plate under certain conditions.