分段Filippov系统的簇发振荡及擦边运动机理

本文旨在揭示非光滑Filippov系统中由频域上不同尺度耦合导致的簇发振荡行为及其产生机理.以经典的周期激励Duffing振子为例,通过引入对状态变量的分段控制及适当选取参数,使得激励频率与系统固有频率之间存在量级差距,建立了频域两尺度耦合的Filippov系统.当激励频率远小于系统的固有频率时,可以将整个激励项视为慢变参数或慢变子系统,从而得到广义自治快子系统.分析了由非光滑分界面划分的不同区域中各快子系统的平衡点及其分岔特性随慢变参数变化的演化过程.考察了两种典型参数条件下系统的振荡行为及其动力学特性,指出参数变化不仅会引起其相应子系统平衡曲线及其分岔特性的改变,也会导致不同模式的簇发振荡.同时,轨迹穿越非光滑分界面时会产生不同的动力学行为,特别是在一定参数条件下,由于运动轨迹受不同子系统的交替控制,存在着擦边运动现象,从而导致特殊形式的非光滑簇发振荡.基于转换相图及各区域中快子系统的平衡曲线及其分岔特性,揭示了非光滑分界面对系统簇发振荡的影响规律及不同簇发振荡的分岔机理.

Bursting oscillations and mechanism of sliding movement in piecewise Filippov system

Since the wide applications in science and engineering, the dynamics of non-smooth system has become one of the key research subjects. Furthermore, the interaction between different scales may result in special movement which can be usually described by the combination of large-amplitude oscillation and small-amplitude one. The influence of multiple scale on the dynamics of non-smooth system has received much attention recently. In this work, we try to explore the bursting oscillations and the mechanism of non-smooth Filippov system coupled by different scales in the frequency domain. Taking the typical periodically excited Duffing's oscillator for example a Filippov system coupled by two scales in the frequency domain is established when the difference in order between the excited frequency and the system natural frequency is obtained by introducing the piecewise control into the state variable and choosing suitable parameters. For the case in which the exciting frequency is far less than the natural frequency, the whole exciting term can be considered as a slow-varying parameter, also called slow subsystem, which leads to a generalized autonomous system, i.e., the fast subsystem. The equilibrium branches and the bifurcations of the fast subsystem along with the variation of the slow-varying parameter in different regions divided according to non-smooth boundary, can be derived. Two typical cases are taken into consideration, in which different distributions of the equilibrium branches and the relevant bifurcations of the fast subsystem may exist. It is pointed out that the variations of the parameters may influence not only the properties of the equilibrium branches, but also the structures of the bursting attractors. Furthermore, since the governing equation alternates between two subsystems located in different regions when the trajectory passes across the non-smooth boundary, the sliding movement along the non-smooth boundary of the trajectory can be observed under the condition of certain parameters. By employing the transformed phase portrait which describes the relationship between the state variable and the slow-varying parameter, the mechanisms of different bursting oscillations and sliding movements are investigated. The results show that bursting oscillations may exist in a non-smooth Filippov system coupled by two scales in the frequency domain. The alternations of the governing equation between different subsystems located in the two neighboring regions along the non-smooth boundary may result in a sliding movement of the trajectory along the non-smooth boundary.