Present paper analyzes the bursting dynamics of a smallest chemical reaction system with multi-frequency slow parametric excitations (referred as SCRSMFSPEs hereafter), in which four novel bursting patterns, i. turnover-of-transcritical-hysteresis-induced bursting, ii. cascaded transcritical-hysteresis bursting, iii. turnover-of-transcritical-hysteresis/supHopf-induced compound bursting and iv. cascaded “supHopf/supHopf” bursting via “delayed transcritical/transcritical” hysteresis loop can be observed. Compared to the original system with a single slow parametric excitation, the hysteresis behavior displays complex dynamical characteristics when an additional parametric excitation is introduced. Typically, the non-zero equilibrium branch, may become the one with twists and turns, leading to more large amplitude oscillations appearing in the active phase of bursting. Based on this, we reveal the novel bursting patterns i and iii. Besides, we show that the non-zero equilibrium branch may cross the zero equilibrium and the supercritical Hopf bifurcation line spirally, resulting in additional transcritical bifurcation points and supercritical Hopf bifurcation points. This gives rise to multiple transcritical-bifurcation-delay-induced hysteresis behaviors, and another two bursting patterns ii and iv are created. Finally, we investigate the transition of these novel bursting, which shows that the transition of bursting is closely related to the excitation amplitudes.

本文分析了具有多频率慢参数激励的最小化学反应系统（以下称为SCRSMFSPE）的爆裂动力学，其中有四个新颖的​​爆裂模式，即。跨临界磁滞引起的周转ii。级联的跨临界磁滞爆裂，iii。跨临界滞后/ supHopf引起的复合爆裂和iv。可以观察到通过“延迟的跨临界/跨临界”磁滞回线的级联“ supHopf / supHopf”爆发。与具有单个缓慢参量激励的原始系统相比，当引入附加参量激励时，磁滞行为显示出复杂的动力学特性。通常，非零平衡分支可能会变成曲折的分支，导致在爆发的活跃阶段出现更大幅度的振荡。基于此，我们揭示了新颖的突发模式i和iii。此外，我们表明非零平衡分支可能会螺旋地越过零平衡和超临界Hopf分叉线，从而导致额外的跨临界分叉点和超临界Hopf分叉点。这引起了多个跨临界分叉延迟引起的磁滞行为，并创建了另外两个突发模式ii和iv。最后，我们研究了这些新型脉冲的跃迁，这表明脉冲的跃迁与激发幅度密切相关。我们表明，非零平衡分支可能会螺旋地越过零平衡和超临界Hopf分叉线，从而产生额外的跨临界分叉点和超临界Hopf分叉点。这引起了多个跨临界分叉延迟引起的磁滞行为，并创建了另外两个突发模式ii和iv。最后，我们研究了这些新型脉冲的跃迁，这表明脉冲的跃迁与激发幅度密切相关。我们表明，非零平衡分支可能会螺旋地越过零平衡和超临界Hopf分叉线，从而产生额外的跨临界分叉点和超临界Hopf分叉点。这引起了多个跨临界分叉延迟引起的磁滞行为，并创建了另外两个突发模式ii和iv。最后，我们研究了这些新型脉冲的跃迁，这表明脉冲的跃迁与激发幅度密切相关。