Fast-slow Behavior Analysis Of Two Classes Of Nonlinear Oscillator With Periodic Excitation

Multiple time-scale dynamics is used to reveal the dynamical laws of nonlinear things, which become one of the frontier and hot problems in the fields of nonlinear dynamics becasue of the wide application of its theories and methods in the fields such as biology, chemistry, life science, geophysics and other scientific fields. In this dissertation, the nonlinear behaviors of two classes of nonlinear oscillator under periodic excitation with two time-scales have been investigated by using the theory of nonlinear dynamics bifurcation, the fast-slow analysis, numercial simulation, et al. The main respects of the research are summarized as follows:A five-dimensional nonlinear fast-slow coupled oscillator which contains an external excitation is investigated. By regarding the term of low-frequency external excitation as a slow variable, a generalized autonomous system is established. The critical condition of two bifurcation models of generalized autonomous system is analyzed, and the bifurcation set as well as the curve of equilibrium points are provided based on bifurcation theory and numerical calculation. With the bifurcation conditions under two sets parameter given before, the periodic bursting oscillation with control parameter is displayed. By using fast-slow analysis and transformed phase, the mechanism of bursting oscillation is discoverd as well as several typical bursting oscillation modes are provided.A five-dimensional nonlinear fast-slow coupled oscillator with two external excitations is discussed. By regarding the two exciting terms as slow variables so that the two slow variables can be expressed in terms of a slow variable with Moivre formula, the original system is divided into fast-slow subsystems. The related critical bifurcation conditions of fast subsystem are analyzed, and the structure of multi-modal oscillation attractors under different excitation frequency ratio are investigated, from which the impact of parity between external excitation frequencies for the system attractor structure is provided. Different multi-modal oscillations with different excitation frequency ratioes are presented, the mechanism of which is obtained via thefast-slow analysis method upon the transformed phase portraits.The coexistence bursting phenomena of a four-dimensional nonlinear fast-slow coupled oscillator which contains an external excitation is investigated. By employing a slow variable so that the exciting term can be expressed in terms of the slow variable,the original system is transformed into a general autonomous system of which equilibrium points as well as related bifurcation conditions are analyzed. The bursting phenomena of periodic exciting system under different initial conditions are presented,the mechanism of which is obtained via the fast-slow analysis method upon the transformed phase portraits. While the coexistence bursting phenomenon of corresponding cluster is discussed, the mechanism of which is obtained via the transformed phase portraits of general autonomous system.