Complex bursting oscillations induced by pulse-shaped explosion and integer frequency ratios in a Rayleigh–van der Pol system

The pulse-shaped explosion (abbreviated as PSE) behavior, represented as the system roots pulse-shaped quantitative changing sharply, is a new way to the bursting oscillations proposed recently. In the present paper, the complex bursting oscillation patterns triggered by the PSE and integer frequency ratios are considered based on a Rayleigh–van der Pol system (abbreviated as RVDPS) excited by the parametrical and external forces. Two cascade bursting oscillations, the cascade supHopf/supHopf bursting oscillations induced by the integer frequency ratios and the cascade supHopf/supHopf bursting oscillations with multiple PSE/PSE hysteresis loops induced by the integer frequency ratios and PSEs, are investigated. We show that the PSEs occur when the system solution curve approaches to infinity at some certain values and the PSEs do not change the stabilities of the attractors. In addition, we find that the frequency and period of the cascade bursting oscillation patterns equal to that of the external force. Our study proposes a new PSE type to the bursting oscillations named the cascade PSEs and increases two cascade bursting oscillation patterns.