High-Dimensional Nonlinear Dynamical Systems And Engineering Applications

The mathematical models and its dynamical equations can be given byhigh-dimensional nonlinear systems in engineering systems. The theoretical method,geometric description and numerical simulation of high-dimensional systems are muchmore sophisticated compare with the low-dimensional nonlinear systems. Therefore, theanalytical methods for the low-dimensional dynamic systems are often unavailable for thehigh-dimensional nonlinear systems. Hence, it is important to provide an all-embracingunderstanding of high-dimensional nonlinear systems. It is a research forefront innonlinear science as well.The existence of the periodic motion for dynamical system is an important theoreticalproblem. There are many studies on these problems and many conclusions have beenobtained for the differential equations of plane autonomous system. The annular regiontheorem, Hopf bifurcations and subharmonic Melnikov method are widely used to solvethese problems, in which the former two methods have been developed to highdimensional system. The subharmonic Melnikov method is rarely used in highdimensional system because of its difficulty in calculating and theoretical analysing.However, this method has been proved to be effcitive to deal with the plane system. In ourstudy, the subharmonic melnikov method is put forward to analyze the existence ofisolation periodic solution under a small perturbation of the parameters in highdimensional system.This dissertation mainly investigates the characteristics of the nonlinear dynamicsand periodic orbits for the high-dimensional nonlinear systems, the primary results includethe following aspects.(1) The period motions of four-dimensional and six-dimensional autonomousnonlinear systems. Using the periodic transformations and Poincaré map, an improvedmethod is presented. Based on the method, we obtain the main theorems, which can beused to analyze the subharmonic nonlinear dynamic responses of high-dimensionalautonomous nonlinear systems, and give a proof using the implicit function theorem. Theperiod motions of the functionally graded material plates and composite laminatedrectangular thin plate are studied by using the improved method.(2) The subharmonic Melnikov method is improved to investigate thefour-dimensional non-autonomous nonlinear dynamical system, and the correctness ofsuch a method is theoretically proved. The periodic motions of a parametrically excitedsimply supported rectangular thin plate in the cases of1:1and1:2internal resonances arestudied by the present method. Numerical simulation is implemented to obtain the peridicmotions of rectangular thin plates. (3) The period-two motion of a simply-supported rectangular honeycomb sandwichplate under the combination of the parametric and transverse excitations is investigated viathe improved four-dimensional subharmonic Melnikov method. The equation of motionfor honeycomb sandwich plate is derived by using the von Karman type equation and theGalerkin approach. We obtained the conditions for the existence of period-two motion bycalculating the Melnikov functions.(4) The subharmonic Melnikov method is improved to investigate thesix-dimensional non-autonomous nonlinear dynamical system. Two theorems are obtainedand can be used to analyze the subharmonic dynamic responses of six-dimensionalnon-autonomous nonlinear systems. The subharmonic Melnikov method was directlyutilized to investigate the subharmonic orbits of a laminated composite piezoelectricrectangular plate in the case of1:2:4internal resonances. The results of numericalsimulation also indicate the existence of the subharmonic orbits for the laminatedcomposite piezoelectric rectangular plate.