We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency.Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples,in which different types of bursting oscillations such as sub Hopf/sub Hopf burster, sub Hopf/fold-cycle burster, and doublefold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states（QSs） and the repetitive spiking states（SPs） may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically.
The main purpose of the paper is to display the relaxation oscillations, known as the bursting phenomena, for the coupled oscillators with periodic excitation with an order gap between the exciting frequency and the natural frequency. For the case when the exciting frequency is much smaller than the natural frequency, different types of bursting oscillations such as fold/fold, Hopf/Hopf bursting oscillations can be observed. By regarding the whole exciting term as a slow-varying parameter on the fact that the exciting term changes on a much smaller time scale, bifurcations sets of the generalized autonomous system is derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two cases with typical bifurcation patterns are focused on as examples to explore the dynamical evolution with the variation of the amplitude of the external excitation. Bursting oscillations which alternate between quiescent states (QSs) and repetitive spiking states (SPs) can be obtained, the mechanism of which is presented by introducing the transformed phase portraits overlapping with the bifurcation diagrams of the generalized autonomous system. It is found that not only the forms of QSs and SPs, but also the bifurcations at the transition points between QSs and SPs, may influence the structures of bursting attractors. Furthermore, the amplitudes and the frequencies related to SPs may depend on the bifurcation patterns from the quiescent sates.
研究多频激励下不同激励频率比时快慢耦合系统的各种复合模态振荡的动力特性及其产生机制.以广义BVP(Bonhoffer-van der Pol)耦合电路为例,通过引入两个电压源控制的电路模块,建立了具有双周期激励的五阶动力模型.选定适当的参数,使得两个激励频率均远小于系统的固有频率,以此考察不同激励频率比下系统的快慢动力学行为.将两个外激励项转化为一个慢变量表达形式,从而将系统分为快慢耦合两子系统.分析了快子系统的平衡点及其分岔条件,探讨了不同激励频率比对复合模态振荡结构的影响,得出系统可能产生原点中心对称,轴对称和非对称的复合模态振荡.给出了系统在6组激励频率下的不同复合模态振荡行为,并进一步揭示了其相应的产生机理.
