Effects of noise on elliptic bursters

Elliptic bursting arises from fast–slow systems and involves recurrent alternation between active phases of large amplitude oscillations and silent phases of small amplitude oscillations. This paper is a geometric analysis of elliptic bursting with and without noise. We first prove the existence of elliptic bursting solutions for a class of fast–slow systems without noise by establishing an invariant region for the return map of the solutions. For noisy elliptic bursters, the bursting patterns depend on random variations associated with delayed bifurcations. We provide an exact formulation of the duration of delay and analyse its distribution. The duration of the delay, and consequently the durations of active and silent phases, is shown to be closely related to the logarithm of the amplitude of the noise. The treatment of noisy delayed bifurcation here is a general theory of delayed bifurcation valid for other systems involving delayed bifurcation as well and is a continuation of the rigorous Shishkova–Neishtadt theory on delayed bifurcation or delay of stability loss.