The pulse-shaped explosion (PSE), characterized by the pulse-shaped quantitative of system solutions varying dramatically, is a special route to bursting oscillations reported recently. This paper reports interesting dynamical behaviors related to the PSE of equilibria, and based on that, the complex bursting dynamics is investigated in a van der Pol–Mathieu–Duffing system with multiple-frequency slow-varying excitations. We find that bifurcations can be observed in a narrow parameter interval within PSE. We also show that two groups of bifurcations are symmetrically arranged on both sides of PSE, and each of which determines a different bursting part. Based on this, two compound bursting patterns, i.e. compound Hopf/Hopf bursting oscillation and compound subHopf/fold cycle bursting oscillation, and a novel type of relaxation oscillation (bursting oscillation of point/point) independent of bifurcations, are revealed. Our results enrich the knowledge of dynamical behaviors related to PSE as well as the possible routes to complex bursting dynamics.

中文翻译：

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van der Pol-Mathieu-Duffing 振荡器中的复杂突发模式

脉冲状爆炸（PSE）以系统解的脉冲状定量变化剧烈为特征，是最近报道的突发振荡的一种特殊途径。本文报道了与平衡 PSE 相关的有趣动力学行为，并在此基础上研究了具有多频慢变激励的 van der Pol-Mathieu-Duffing 系统中的复杂爆发动力学。我们发现可以在 PSE 内的狭窄参数区间内观察到分叉。我们还展示了两组分叉对称地排列在 PSE 的两侧，并且每一个分叉决定了不同的爆裂部分。在此基础上，产生了两种复合爆发模式，即复合 Hopf/Hopf 爆发振荡和复合 subHopf/fold 循环爆发振荡，并揭示了一种与分叉无关的新型弛豫振荡（点/点的爆发振荡）。我们的结果丰富了与 PSE 相关的动态行为的知识，以及复杂突发动态的可能途径。