The Complicated Dynamics In Several Classes Of High Dimensional Nonlinear Engineering Systems

As modern technology and engineering develop and require, simplification ofengineering systems based on linearization which neglects high ordered nonlinearitiescan no longer accurately predict the dynamics, nor give explanations for somenonlinear phenomena of engineering and physic, such as limit circles, bifurcations andchaos. A variety of engineering and physical problems must be given by highdimensional nonlinear systems, so related nonlinear theories must be developed andapplied to real world to reveal the complex dynamics in the high dimensionalnonlinear systems.Theories for nonlinear systems, especially for high dimensional nonlinearsystems, are very rare. As one of the few analytical methods, the extended Melnikovmethod is a powerful tool in examining the chaotic behavior induced by homoclinic orheteroclinic tangles, and determining the parametric intervals for chaos occurring.Combined with geometric singular perturbation theory, the method enables us to findthe conditions for the existence of the Shilnikov-type multi-pulse orbit in the fourphase space under resonance. The existence of the orbit implies a horseshoe chaos inthe system.In this dissertation, the nonlinear dynamics is studied in two classes ofengineering systems, a cantilever plate system in macro scale and a nano oscillatorsystem, to investigate the multi-pulse chaotic motion. Analytical study on the globaldynamics of an engineering system is rare, partly because the theories are hard to find,and partly due to the demanding prerequisite for applying the few analytical methodsto engineering systems. Based on the nonlinear governing equation of the engineeringsystems, the paper employs coordinate transformation to obtain four-dimensional nearintegrable Hamiltonian systems which the extended Melnikov method can be used to.This system is close to the original system and the results conducted from it havevalue for engineering applications. The paper can be divided into the followingchapters.(1) The dynamics of the unperturbed system of the composite cantilever platemodel. The nonlinear governing equations for the first and second modes of acomposite cantilever plate are given. The perturbed four-dimensional Hamiltoniansystem is obtain through the method of multiple scales and coordinate transformation.The dynamics on the unperturbed system is given.(2) Global bifurcations and multi-pulse chaotic dynamics of the compositecantilever plate system. For the perturbed system, the parametrical conditions of chaosfor the resonance case are given. The numerical simulations include phase portraits, wave forms and power spectrums, which verify the theoretical analysis.(3) The dynamics of the unperturbed system of a nonlinear nano-resonator withcubic nonlinearity and parametrical excitations. The nonlinear governing equationsfor a coupled nano-oscillator are given based on a nano-plate model. The dynamics onthe unperturbed system is given.(4) The Shilnikov multi-pulse orbit and chaotic dynamics of the nano-resonatorsystem. The Shilnikov-type multi-pulse homoclinic orbit in the nano-plate system isanalyzed. The numerical analysis shows the relation among phase angle, dissipativefactor, detuning parameter and the pulse number. The phase portraits and wave formsare given to demonstrate the existence of the Shilnikov-type multi-pulse homoclinicorbit.