Delayed bifurcation behavior is ubiquitous in multi-time scale systems and has a great important effect on bursting oscillations. In this paper, a parametrically driven Shimizu-Morioka system is proposed and distinct delay behavior is observed when the slow-varying parameter passes through the transcritical bifurcation point periodically. Periodic and chaotic bursting oscillations induced by such delay behavior are investigated by using the fast-slow analysis method with different excitation amplitudes. More interesting, the fast subsystem has a zero equilibrium point branch and two asymmetrically distributed equilibrium point branches with respect to the slow-varying parameter. Thus, there exist two possible routes for the trajectory to choose when the delayed transcritical bifurcation takes place. This leads to that the bursting patterns corresponding to different excitation periods may be different. As a result, two possible mixed bursting patterns, named as “delayed transcritical/transcritical” bursting of point-point type mixed with “delayed transcritical/supHopf/fold” bursting of point-cycle type and “delayed transcritical/subHopf/subHopf/fold” bursting of point-cycle type mixed with “delayed transcritical/supHopf/fold” bursting of point-cycle, are revealed. Furthermore, the effect of the excitation frequency on the delay behavior is also considered. We find the delay behavior may terminate at different parameter areas when the excitation amplitude is fixed and the excitation frequency varies and thus leads to different bursting patterns. Numerical simulations are provided to verify the validity of the study.

延迟分叉行为在多时间尺度系统中无处不在，并且对突发振荡具有重要的影响。本文提出了一种参数驱动的Shimizu-Morioka系统，当慢变参数周期性地通过跨临界点时，观察到明显的延迟行为。通过使用具有不同激励幅度的快速慢速分析方法，研究了由这种延迟行为引起的周期性和混沌突发振荡。更有趣的是，相对于慢速变化参数，快速子系统具有零平衡点分支和两个非对称分布的平衡点分支。因此，当发生延迟的跨临界分叉时，存在两种可能的路径供轨迹选择。这导致对应于不同激励时段的突发模式可以不同。结果，出现了两种可能的混合突发模式，即点-点类型的“延迟跨临界/跨临界”突发与点循环类型的“延迟跨临界/ supHopf / fold延迟”混合和“延迟跨临界/ subHopf / subHopf / fold延迟”混合揭示了点循环类型的“突发”与“延迟的跨临界/ supHopf /折叠”点循环的混合。此外，还考虑了激励频率对延迟行为的影响。我们发现，当激励幅度固定且激励频率变化时，延迟行为可能会在不同的参数区域终止，从而导致不同的突发模式。提供数值模拟以验证研究的有效性。