Quasi-Periodicity at Transition from Spiking to Bursting in the Pernarowski Model of Pancreatic Beta Cells

Abstract

This paper studies quasi-periodicity phenomena appearing at the transition from spiking to bursting activities in the Pernarowski model of pancreatic beta cells. Continuing the parameter, we show that the torus bifurcation is responsible for the transition between spiking and bursting. Our investigation involves different torus bifurcations, such as supercritical torus bifurcation, saddle torus canard, resonant torus, self-similar torus fractals, and torus destruction. These bifurcations give rise to complex or multistable dynamics. Despite being a dissipative system, the model still exhibits KAM tori, as we have illustrated. We provide two scenarios for the onset of resonant tori using the Poincaré return map, where global bifurcations happen because of the saddle-node or inverse period-doubling bifurcations. The blue-sky catastrophe takes place at the transition route from bursting to spiking.