Undetermined Fundamental Frequency And The Complicated Dynamics Of Nonlinear Dynamical System

General nonlinear dynamical systems contain the complicated dynamical phenomenon mostly, such as bifurcation and chaos. That makes it the fundamental motility for the emerging, developing, and gradually optimizing of the modern nonlinear dynamical research because the complicated problems exciting and unfamiliar phenomenon rush out in the physical world all the time. The emergency of the undetermined fundamental frequency just accorded with the circumstance, where the research of the normal form theory has come to its mature stage and the strongly nonlinear problems bring us the enough temptation to practise our research. In this article, the initial targets of the method is extend, from pursuiting the steady asymptotic responses to the broader areas. It includes the static and dynamic bifurcation behaviors of the strongly nonlinear oscillation system (SNOS), improving the computational precision of the Homoclinic and Heteroclinic bifurcation in terms of the Melnikov method, and Shilnikov sense Homoclinic orbit and flip (periodic doubling) bifurcation in the three dimensional nonlinear system. So we pay our attention to these fields with the application of modern nonlinear dynamical methodology, to find the appropriate resolvents.The research contents and obtained major results of this dissertation are as follows.(1). The analytical criterions of the Homoclinic (Heteroclinic) bifurcation of the strongly nonlinear oscillator are presented. We consider the approximate periodic solution of the system subject to the quintic nonlinearity by introducing the undetermined fundamental frequency into the complex normal form operation. For the occurrence of Homoclinicity (Heteroclinicity), the bifurcation criterions are accomplished. They depend on the contact of the limit cycle with the saddle equilibrium and the vanishing undermined fundamental frequency. So the available ranges of the former criterions are extended from the weakly nonlinear oscillation system to the SNOS.(2). The static bifurcation of the parametrically excited rotating arm with strongly nonlinearity is researched. For the discussion of static bifurcation, the bifurcation problem is described as the 3-codimension unfolding with Z2 symmetry on the basis of the singularity theory. The transition set and bifurcation diagrams, for the singularity are wholly presented with constraints, while the stability of the zero solution is researched by the eigenvalues in various parameter regions.(3). The dynamic bifurcations of strongly nonlinear oscillator induced by parametric and external excitation are researched. It is known that the parametric and external excitation may induce additional saddle states, and result chaos in the phase space, which can not be detected by applying the Melnikov method directly. A feasible solution for this problem is the combination of the averaged equations and Melnikov method. So we consider the averaged equations of the system subject to Duffing-Van der Pol type strong nonlinearity by introducing the undetermined fundamental frequency. Then the saddle states and bifurcation values of Homoclinic structure formation are detected through the combined application of the new averaged equations with Melnikov method.(4). The simple approach to improve the computational precision of Melnikov method is presented by using the undetermined fundamental frequency and normal form method. We construct the improved Melnikov expression for a triple well nonlinear oscillator subject to principal parametric resonance and external excitation. For the occurrence of chaos, the threshold value approximations of chaotic motion are obtained in the Homoclinicity and Heteroclinicity points of view. It depends on the introduction of undetermined fundamental frequency, and adopting new time transformation for fulfilling the Homoclinic and Heteroclinic orbits, so that the effect of disturbing parameter can be easily detected and embodied in the Melnikov operation.(5). The strategy of predicting the Shilnikov sense horseshoe and Homoclinic orbit of the new PID controller and reduced solar wind driven magnetosphere ionosphere model (WINDMI) is presented by using the Shilnikov theorem and the invariant manifolds of the saddle-focus equilibrium point. It provides the quadratic and cubic nonlinearity to the controller systems, and extends the canonical forms of continuous time quadratic autonomous chaotic systems in three dimensions. The Shilnikov type Homoclinic orbit is concerned not only about its existence but also the analytical series expression, which distinguishes the present work from the general description of the Shilnikov sense chaos. For the discussion of flip bifurcation in the three dimensional system, the introduction of the undetermined fundemental frequency improve the computational precise of normal form method and efficiently avoid the difficulties of seeking higher-order asymptotic solutions during the course of operation.