多尺度耦合非线性动力系统的复杂行为及其机理研究

多尺度耦合非线性动力系统不仅具有广泛的工程背景，也会存在特殊的动力学行为，成为当前国内外研究的热点和前沿课题之一。本项目紧密围绕不同尺度耦合这一主题，首先探讨时域两尺度耦合也即自治系统情形，分析不同快慢子系统维数下簇发振荡行为，建立多慢变量的分析方法，重点考察多种沉寂态和激发态共存、多种分岔模式共存及高余维分岔下各种簇发振荡行为及其产生机制，然后探讨不同周期激励下频域两尺度情形，提出频域两尺度耦合的分析方法，分析其不同簇发振荡及其产生机制，给出与时域两尺度情形的簇发振荡之间的区别及其原因，进而推广到三时间尺度上。结合簇发振荡中分岔标准型及其相应的普适开折，并在开折参数意义下将相关因素分类，给出相应的拓扑结构，从而得到不同簇发的标准型，揭示其簇发振荡的本质及其机理，初步构建以快慢子系统维数为纲目的多尺度耦合非线性动力系统的理论框架体系，为解决实际工程问题提供理论支撑。

The nonlinear dynamical systems with multiple scales not only have a wide range of engineering background, but also may appear in special dynamical behaviors, the characteristics of becomes one of the hot and frontier issues in the research area of nonlinear dynamics at home and abroad. In the project, focused on the subject with coupling of multiple scales, we will first investigate the autonomous systems with two scales in time domain, and explore the bursting oscillations in the dynamical systems with different dimensions of the fast and slow subsystems. Based on the establishment of the analytical method for multiple slow variables, different bursting oscillations as well as the mechanism will be presented, especially for the cases with the coexistence of multiple forms of quiescent states and repetitive spiking states, the coexistence of multiple bifurcation forms and the existence of high co-dimensional bifurcations. Then we turn to the dynamical systems with different forms of periodic excitations with multiple scales in frequency domain. The analytical method will be presented, based on which different types of bursting oscillations as well as the mechanism will be presented. The difference as well as the reason between the bursting oscillations in time and frequency domain, respectively, will be discussed. All the results will be extended to the systems with three time scales. Combining with the normal forms of the bifurcations in the bursting oscillations as well as their universal unfolding expressions, all the associated factors will be classified into different groups according to the unfolding parameters. Upon the corresponding topological structures, the norm forms of different bursting oscillations will be presented, which can be employed to explore the characteristics of bursting oscillations as well as the mechanism. The preliminary systematical theory of nonlinear dynamics with multiple scales will be established according to the dimensions of slow and fast subsystems, which may supplies the theoretical support to solve the engineering problems with multiple scales.