Complex Dynamics Analysis In Coupled Nonlinear Systems

The dynamical analysis of high dimensional coupled nonlinear systems is one of the frontier problems in the study of nonlinear dynamical systems. Based on the results we've obtained, the mechanism of the complexity is discussed by applying the modern nonlinear analytical methods to the systems. The details of the influence on the dynamical behavior caused by the physical parameters and the initial conditions are also explored in this thesis in order to reveal the quality of the complexity, which may provide the theoretical basement for solving practical engineering problems, such as nonlinear modeling, parametrical identification and failure diagnose.Firstly, the dynamical behaviors of the shallow arch, possessing initial static deformation with periodic parametric excitation, under the internal resonance circumstance are analyzed in this thesis. According to the stability criteria, the physical parametric space is divided into different regions, associated with different types of steady state solutions, which are proved by employing numerical methods. A route to chaos is found in the evolving process of the movements of the system.Secondly, the 0:1 internal resonant case in the shallow arch system is taken into consideration. The heterclinic orbit bifurcations are studied in detail by applying the global perturbation method developed by Kovacic and Wiggins. The effect caused by the unfolding parameters of the second order in the perturbed system are considered while computing the high dimensional Melnikov function. The necessary conditions of the chaos break-out in Silnikov type homoclinic orbits with Smale's definition are obtained, together with more ways of bifurcation.Thirdly, a universal triple mechanical model in engineering problem is discussed in this thesis. Considering the influence of the nonlinear damping, the different steady state solutions and their stability are analyzed after applying with the center manifold theorem and the normal form theory of the vector fields whose characteristic polynomial has three pairs of pure imagine eigenvalues without resonance. Two different evolving ways from equilibrium to high dimension torus are pointed out.In addition, the numerical methods are used to verify the reliability and effectiveness of the theoretical analysis of the three different models respectively and some meaningful results are summarized in the end of this thesis. Also some existing problems as well as the future work are pointed out.