Quasi-periodic solutions and periodic bursters in quasiperiodically driven oscillators

In this paper, we propose a perturbation method to determine an approximation and conditions of existence of quasi-periodic (QP) solutions and bursting dynamics in a quasi-periodically driven system. The QP forcing consists of two periodic excitations, one with a very slow frequency and the other with a frequency of the same order of the proper frequency of the oscillator. A first averaging is done over the fast dynamics, then the quasi-static solutions of the modulation equations of amplitude and phase are determined and their stability analyzed. We show that a necessary condition for the occurrence of periodic bursters is that the slow excitation is parametric.