共存吸引子对簇发振荡影响机理分析

Influence mechanism of coexisted attractors on bursting oscillations

具有广泛工程和科学背景的多稳态系统存在着多个共存的吸引子及其相应的吸引域,其不同尺度下的行为会受到共存吸引子的影响而产生复杂的动力学特性,该文章旨在探讨共存吸引子下簇发振荡的特性及其产生机制。基于经典的Hartley振子,引入周期外激,当激励频率远小于系统的固有频率时,随着参数的变化,系统呈现出不同簇发振荡模式。将周期激励项视为慢变参数,基于相应的广义自治快子系统的平衡点及其稳定性分析,得到其随慢变参数变化的平衡曲线及其相应的分岔图。发现在不同参数条件下,存在不同类型的多稳态吸引子共存情形。考虑3种典型情形下系统的快慢效应,分析各情形下共存稳态吸引子的特性及吸引域对分岔特性的影响,得到连接沉寂态和激发态的分岔模式,从而揭示相应的簇发振荡机理。指出当轨迹依次穿越不同共存吸引子的吸引域时,各稳定吸引子均会对簇发振荡的结构产生影响,使得簇发振荡变得复杂,而当轨迹仅穿越部分共存吸引子的吸引域时,根据初始条件的不同,会产生共存的簇发振荡。

There exist coexisted stable attractors with different attracting basins in multi-stable systems with wide science and engineering background,and complicated dynamical behaviors of the system under the coupling of different scales can be observed under the influence of the attractors.The paper aims at the effect of the coexisted stable attractors on the property of bursting oscillations as well as its mechanism in the multi-stable systems.Taking a classical Hatley oscillator as an example,by introducing an external periodic excitation,with the frequency far less than the natural frequency,different forms of bursting oscillations were obtained along with the variation of parameters.Regarding the whole exciting term as a slow-varying parameter,the equilibrium branches and their bifurcations along with the variation of the slow-varying parameter were derived based on the stability analysis of the equilibrium point of corresponding generalized autonomous fast subsystem.It is found that there exist several coexisting situations of different types of coexisted stable attractors.Here three typical cases were derived based on where the characteristics of the coexisted attractors and the influence of their attracting basins on the bifurcation were analysed.Accordingly,the bifurcation modes at the transitions between the quiescent states and spiking states were obtained,which can be used to explore the mechanism of the bursting oscillations.It is pointed out that when the trajectory passes across the attracting basins of the coexisted attractors in turn,the associated attractors may affect the structure of the bursting oscillations,which leads to complicated bursting oscillations.However,when the trajectory only passes across the attracting basins of some of the stable attractors,there may exist coexisted bursting attractors,which correspond to different initial conditions.