Bursting oscillations in Sprott B system with multi-frequency slow excitations: two novel “Hopf/Hopf”-hysteresis-induced bursting and complex AMB rhythms

Abstract

Bursting oscillations in systems with multi-frequency slow excitations reflect multi-timescale dynamics. As there exist many kinds of multiple periodic excitations in actual dynamical systems, the analysis of dynamical behavior of such systems has received much attention. In this article, we first investigate the bursting in Sprott B system with a single excitation. We show that Hopf bifurcation delay may exhibit due to the effect of slow passage through the supercritical Hopf bifurcation. Interestingly, such delay behavior leads to a pinched “Hopf/Hopf”-hysteresis bursting when delay time is long enough. Then, we focus on the effects of the additional excitation $${\beta _2}{\mathrm{cos(}}{\omega _2}{\mathrm{t)}}$$ on the zero-crossing pinched “Hopf/Hopf”-hysteresis bursting. We find that, under the effects of $${\beta _2}{\mathrm{cos(}}{\omega _2}{\mathrm{t)}}$$ , “Hopf/Hopf”-bifurcation-delay-induced hysteresis behaviors may exhibit very interesting dynamical characteristics. In particular, each equilibrium branch may become the one with twists and turns. Then, the novel bursting pattern, turnover-of-zero-crossing pinched-“Hopf/Hopf”-hysteresis-induced bursting, is disclosed accordingly. In addition, we show that the equilibrium branches may cross each other spirally, leading to multiple intersection points under certain parameter conditions. There are two Hopf bifurcation points perching separately upon either side of the intersection points. This creates multiple delay-induced hysteresis behavior, and the other novel bursting pattern, cascaded pinched “Hopf/Hopf”-hysteresis bursting, is thus created. Finally, we investigate the complex amplitude-modulated bursting (CAMB) rhythms in Sprott B system with multi-frequency slow excitations.