Sub-Hopf/fold-cycle bursting and its relation to (quasi-)periodic oscillations

We investigate the emergence of bursting oscillations and their relation to (quasi-) periodic behaviour in two different model systems, a pH oscillator and a calcium oscillator. Both systems are described by 3-dimensional ODE systems and follow different ‘routes’ to bursting oscillations as parameters are varied. In the first part, we exploit the slow-fast structure of the 3-dimensional ODE systems to show that the bursting oscillations are of sub-Hopf/fold-cycle type over a wide range of parameters. For this purpose, a slow variable is used as a quasi-static bifurcation parameter for the remaining 2-dimensional fast subsystem. For the pH oscillator, a subsequent two-parameter continuation reveals a transition in the bursting behaviour from sub-Hopf/fold-cycle to fold/sub-Hopf type. In the second part, we argue that the particular constellation of a subcritical Hopf bifurcation together with a saddle-node bifurcation of a periodic orbit (fold-cycle) may not only account for the bursting oscillations, but also allows for (quasi-)periodic oscillations on a 2-torus suggesting a common origin for both types of dynamics.