Complex bursting patterns and fast-slow analysis in a smallest chemical reaction system with two slow parametric excitations

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.