串联式叉型滞后簇发振荡及其动力学机制

SERIES-MODE PITCHFORK-HYSTERESIS BURSTING OSCILLATIONS AND THEIR DYNAMICAL MECHANISMS

簇发振荡是自然界和科学技术中广泛存在的快慢动力学现象,其具有与通常的振荡显著不同的特性.根据不同的动力学机制可将其分为多种模式,例如,"点–点"型簇发振荡和"点–环"型簇发振荡等.叉型滞后簇发振荡是由延迟叉型分岔诱发的一类具有简单动力学特性的"点–点"型簇发振荡.研究以多频参数激励Duffing系统为例,旨在揭示一类与延迟叉型分岔相关的具有复杂动力学特性的簇发振荡,即串联式叉型滞后簇发振荡.考虑了一个参激频率是另一个的整倍数情形,利用"频率转换快慢分析法"得到了多频参数激励Duffing系统的快子系统和慢变量,分析了快子系统的分岔行为.研究结果表明,快子系统可以产生两个甚至多个叉型分岔点;当慢变量穿越这些叉型分岔点时,形成了两个或多个叉型滞后簇发振荡;这些簇发振荡首尾相接,最终构成了所谓的串联式叉型滞后簇发振荡.此外,分析了参数对串联式叉型滞后簇发振荡的影响.

Bursting oscillations is a spontaneous physical phenomenon existing in natural science,which has various patterns according to their dynamical regimes.For instance,bursting of point-point type means bursting patterns related to transition behaviors among di erent equilibrium attractors.Pitchfork-hysteresis bursting,induced by delayed pitchfork bifurcation,is a kind of point-point type bursting pattern showing simple dynamical characteristics.The present paper takes the Duffng system with multiple-frequency parametric excitations as an example in order to reveal bursting patterns,related to delayed pitchfork bifurcation,showing complex characteristics,i.e.,the series-mode pitchfork-hysteresis bursting oscillations.We considered the case when one excitation frequency is an integer multiple of the other,obtained the fast subsystem and the slow variable of the Duffing system by frequency-transformation fast-slow analysis,and analyzed bifurcation behaviors of the fast subsystem.Our study shows that two or multiple pitchfork bifurcation points can be observed in the fast subsystem,and thus two or multiple pitchfork-hysteresis bursting patterns are created when the slow variable passes through these points.In particular,the pitchfork-hysteresis bursting patterns are connected in series,and as a result,the so-called series-mode pitchfork-hysteresis bursting oscillations are generated.Besides,the effects of parameters on the series-mode pitchfork-hysteresis bursting oscillations are analyzed.It is found that the damping of the system and the maximum excitation amplitude show no qualitative impact on corresponding dynamical mechanisms,while the smaller one may lead to vanish of busting oscillations.Our findings reveal the road from simple dynamical characteristics of point-point type bursting oscillation related to complex one,thereout,a complement and expansion for nowadays bursting dynamics is obtained.