Bursting multistability induced by double-Hopf bifurcation

Abstract

We study the slow–fast dynamics of a system with a double-Hopf bifurcation and a slowly varying parameter. The model consists of coupled Bonhöffer–van der Pol oscillators excited by a periodic slow-varying AC source. We consider two cases where the slowly varying parameter passes by or crosses the double-Hopf bifurcation, respectively. Due to the system’s multistability, two bursting solutions are observed in each case: single-mode bursting and two-mode bursting. Further investigation reveals that the double-Hopf bifurcation causes a stable coexistence of these two bursting solutions. The mechanism of such coexistence is explained using the slowly changing phase portraits of the fast subsystem. We also show the robustness of the observed effect in the vicinity of the double-Hopf bifurcation.