频域两尺度下非光滑Duffing系统的簇发振荡及其机理分析

Bursting oscillations and mechanism analysis in a non-smooth Duffing system with frequency domain of two time scales

为揭示非光滑因素存在时动力系统的簇发行为,本文以周期激励Duffing系统为例,引入干摩擦,同时选取周期激励频率与系统的固有频率之间存在量级差距,构建了频域两尺度下非光滑动力系统.将周期激励项视为慢变参数,激励系统转化为广义自治系统,考察了广义自治系统平衡点的稳定性以及分岔条件.选取适当的参数值,重点研究了三组参数条件下的簇发现象,即双涡卷情形、三涡卷情形和四涡卷情形.结合转换相图,并考虑到非光滑因素的影响,讨论了相应快子系统的分岔模式,揭示了该类非光滑系统中不同簇发振荡的产生及其沉寂态与激发态之间相互转迁的分岔机理.另外,针对于该类非光滑系统中的多平衡态共存现象,结合吸引子自身的演化行为对涡卷形成的影响,进一步揭示了系统产生特殊振荡现象的原因.

The main purpose of the manuscript is to explore the bursting oscillations in a dynamic system with non-smooth factor. Take the duffing system as a example, by introducing dry friction, and choosing periodic excitation ensure that there exists order gap between the natural frequency and the exited frequency, a non-smooth system with two time scales is established. The whole exciting term can be regarded as a slow-varying parameter, which leads to the generalized autonomous system, and the stability and the associated bifurcation conditions can be derived. By taking suitable parameter values, three typical cases are considered under the three group different parameters which can described as double scrolls, three scrolls and four scrolls. By using the transformed phase portrait and considering the effect of non- smooth factor, the associated bifurcation mechanism of the bursting oscillations is discussed, which may reveal different forms of bursting oscillations and transition mechanism between quiescent states and the spiking states. In addition, aiming at the coexistence of multiple equilibrium states in this non-smooth system, combined with the influence of evolution behavior of the attractor itself, and further reveals that the reason for showing special oscillation phenomena under the influence of non-smooth factor.