Studies On Global Bifurcations And Multi-Pulse Chaotic Dynamics Of High-Dimensional Nonlinear Systems And Its Applications

The mechanical models for a variety of problems in the field of mechanics, aircraft, aerospace, mechanical engineering, can be described by high-dimensional nonlinear systems. When these systems are subjected to external excitations and internal coupling of nonlinearity, complex dynamical behaviors of the high-dimensional nonlinear systems will occur, such as modal interactions, the transfer of energy, the phenomena of jumping, multi-pulse chaotic dynamics. It is a meaningful topic to find the mechanism of these complex dynamical behaviors and improve reasonable control on these systems. Recently, studies on complex dynamics in high-dimensional nonlinear systems have become the leading, significant and difficult topics in the field of nonlinear dynamics.Up to now, there is very few analytical method which can be used to study the global dynamics of high-dimensional nonlinear systems. These methods mainly include the Melnikov method, the global perturbation method for investigating the single-pulse chaotic dynamics, the extended Melnikov method and energy-phase method for studying the multi-pulse chaotic dynamics in the case of resonance, and the method of exponential dichotomies for studying the chaotic dynamics in the case of non-resonance. These methods are powerful tools to investigate the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering. However, these methods have many limitations for applying them to analyze the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering. For example, the unperturbed parts of high-dimensional nonlinear systems, which can be analyzed by the energy-phase method, are generally requested to be four-dimensional Hamiltonian systems. The extended Melnikov is usually employed to analyze the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems with the action-angle variable. The unperturbed part of the action-angle variable may not be a Hamiltonian system. However, other parts of the unperturbed systems must be Hamiltonian with completely integrability. The method of exponential dichotomies can be utilized to analyze the chaotic dynamics of high-dimensional nonlinear systems. However, the process of analysis is too complex and lack of the geometric intuition for engineering researchers.In this dissertation, firstly, the extended Melnikov method is further generalized to study the global bifurcations and multi-pulse chaotic dynamics for a class of high-dimensional nonlinear systems. The energy-phase function is derived for the four-dimensional subsystem of a class of high-dimensional nonlinear systems. The Melnikov method is also improved to study the global bifurcations and chaotic dynamics for a class of high-dimensional non-autonomous slowly varying oscillators. Then, the aforementioned global perturbation methods are utilized to investigate the global bifurcations and multi-pulse chaotic dynamics of several engineering systems. The results obtained here reveal the mechanism on complex nonlinear dynamical behaviors and the instability of motions, such as the multi-pulse jumping and multi-pulse chaotic motions of high-dimensional nonlinear systems in engineering.The major research contents of this dissertation are briefly summarized as follows.(1) After improving the limitations of completely integrability for the Melnikov method in high-dimensional nonlinear systems and using the structure of solutions for variational equation along homoclinic manifolds, the theory of fibers for invariant manifolds and geometric singular perturbation theory, the extended Melnikov method is further improved to study the global bifurcations and multi-pulse chaotic dynamics for a class of high-dimensional nonlinear systems.(2) A class of four dimensional nonlinear systems subjected to both Hamiltonian and dissipative perturbations are investigated. The energy-phase function for N-pulse jumping orbits is derived using the extended Melnikov function for the first time. The energy-phase function is the same as the one given by Haller in energy-phase method when the unperturbed system is Hamiltonian. Moreover, the differences and connections between the energy-phase method and the extended Melnikov method are also analyzed.(3) Applying the technique of fast manifold and the structure of solutions for variational equation along homoclinic manifolds, the Melnikov method is improved to investigate the global bifurcations and chaotic dynamics for a class of high-dimensional non-autonomous slowly varying oscillators. The generalized methods are more suitable to analyze the global dynamics of complex engineering systems.(4) Based on the governing equations of nonlinear nonplanar motion for a cantilever beam, the global bifurcations and multi-pulse chaotic motions for the cantilever beam in the case of one-to-one internal resonance are analyzed using the extended Melnikov method. The global bifurcations and multi-pulse chaotic motions for the cantilever beam in the case of one-to-two internal resonance are investigated using the energy-phase method for the first time. These results obtained here can reveal the mechanism of complex nonlinear motions for the cantilever beam, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are implemented to demonstrate the existence of the complex nonlinear phenomena, such as multi-pulse jumping and the multi-pulse chaotic motions for the cantilever beam.(5) The exponential dichotomies and the generalized averaged procedure are employed to study the chaotic motions for a buckled rectangular thin plate for the first time. In the case of primary resonance and principle parametric resonance, it is found that non-hyperbolic subsystem of the rectangular thin plate does not affect the critical condition on the occurrence of chaotic motions for the full system of the rectangular thin plate. Numerical simulations are implemented to explain the influence of external and parametric excitations on the chaotic motions for the rectangular thin plate. Numerical results show that the chaotic motions of the hyperbolic subsystem are shadowed by the chaotic motions for the full system of the rectangular thin plate.(6) The global bifurcations and multi-pulse chaotic dynamics for a functionally graded material rectangular plate are investigated in the case of one-to-one internal resonance using the extended Melnikov method. In order to overcome the difficulty for dynamical analysis brought by the square and cubic nonlinear terms in the governing equations of motion, the method of high-order multiple scales is applied to derive the averaged equation governing the amplitude and phase. The extended Melnikov method is employed to detect the mechanism of complex motions for the FGM rectangular plate, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are finished to study the influence of the external and parametric excitations on the nonlinear dynamics for the FGM rectangular thin plate. Numerical results indicate the existence of multi-pulse chaotic motions and other complex nonlinear phenomena for the FGM rectangular plate.(7) The global bifurcations and multi-pulse chaotic dynamics for a nonlinear vibration absorber are studied in the case of one-to-one internal resonance using the energy-phase method. The mechanism of complex motions for the nonlinear vibration absorber is detected, such as modal interactions, the transfer of energy, the multi-pulse jumping and the multi-pulse chaotic motions. Numerical simulations are given to study the effect of the parametric excitation, the damping parameter and the detuning parameters on the nonlinear dynamic responses for the nonlinear vibration absorber. Numerical results illustrate the existence of multi-pulse jumping, multi-pulse chaotic motions and other complex nonlinear phenomena for the nonlinear vibration absorber.In the last section, the dissertation is summarized. Moreover, the further studies on global bifurcations and multi-pulse chaotic dynamics for high-dimensional nonlinear systems are discussed.