周期激励下四维非线性系统的簇发共存现象

BURSTING COEXISTENCE PHENOMENON OF A FOUR-DIMENSIONAL NON-AUTONOMOUS SYSTEM UNDER PERIODIC EXCITATION

在一个周期激励的四维非自治系统中,当激励的频率远小于系统的固有频率时,系统表现出了两时间尺度的动力学行为.将激励项定义为慢变参数,激励系统可以转化为广义自治系统.分析了广义自治系统平衡点的稳定性及其分岔条件.应用快慢分析法和转换相图,探讨了系统对应于不同初始条件的簇发现象及其产生机制,并对其中多种簇发共存的形成机理进行了讨论.同时,由于慢过效应的存在,簇发振荡的激发态和沉寂态的连接点和理论分析中的分岔点相比存在一定的滞后现象.

For a periodically excited four-dimensional non-autonomous system, when the exciting frequency is much less than its nature frequency, dynamical behaviors associated with two time scales can be observed. The excited system can be transformed into a general autonomous system by defining the whole exciting term as a slow- varying parameter. Firstly, the stability and bifurcation conditions of equilibrium points in the generalized autono- mous system are presented. Secondly, the slow-fast analysis and transformed phase are employed to explore the different types of bursting behaviors with different initial conditions. In addition, the mechanism of coexistence phenomenon is discussed. Meanwhile, delay phenomenon is observed between the points connecting the spiking states and the quiescent states and the bifurcation points obtained theoretically.