中文摘要：

旨在揭示频域不同尺度耦合时非对称动力系统簇发振荡的特点及其分岔机理，并进一步揭示快子系统多平衡点共存导致的不同簇发模式及其产生原因．以经典的蔡氏振子为例，通过引入非对称控帝1．项及周期变化的电流源．选取适当参数，构建存在频域两尺度耦合的非对称动力系统模型．当周期激励频率远小于系统的固有频率时，将整个周期激励项视为慢变参数，得到随慢变参数变化的快子系统平衡曲线及其不同的分岔点以及分岔行为．重点分析了三种不同周期激励幅值下典型的非对称簇发振荡及吸引子结构，揭示其相应的产生机理．指出外激励幅值的变化不仅会引起不同稳定平衡点吸引域的变化，也会使得慢变量穿越不同分岔点的时间间隔发生变化，导致系统产生不同形式的簇发振荡．

英文摘要：

The main purpose of this study is to investigate the characteristics as well as the bifurcation mechanisms of the bursting oscillations in the asymmetrical dynamical system with two scales in the frequency domain. Since the slow-fast Hodgkin-Huxley model was established to successfully reproduce the activities of neuron, the complicated dynamics of the system with multiple time scales has become a hot research topic due to the wide engineering background. The dynamical system with multiple scales often presents periodic oscillations coupled by large-amplitude oscillations at spiking states and small-amplitude oscillations at quiescent states, which are connected by bifurcations. Up to now, most of the reports concentrate on bursting oscillations in the symmetric systems, in which there exists only one form of spiking oscillations and quiescence, respectively. Here we explore some typical forms of bursting behavior in an asymmetrical dynamical system with periodic excitation, in which there exists an order gap between the exciting frequency and the natural frequency. As an example, based on the typical Chua＇s oscillator, by introducing an asymmetrical controller and a periodically .changed current source, and choosing suitable parameter values, we establish an asymmetrical dynamical system with two scales in the frequency domain. Since the exciting frequency is much smaller than the natural frequency, the whole periodic exciting term can be regarded as a slowly-varying parameter, leading to the fast subsystem in autonomous form. Since all the equilibrium curves and relevant bifurcations are presented in the form related to the slowly-varying parameter, the transformed phase portraits describing the evolution relationship between the state variables and the slowly-varying parameter are employed to account for the mechanism of the bursting oscillations. With the variation of the slowly-varying parameter, different equilibrium states and relevant bifurcations in the fast subsystem are presented. It is found that fo