Mixed-mode oscillations in three time-scale systems: A prototypical example

Mixed-mode dynamics is a complex type of dynamical behavior that is characterized by a combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode oscillations (MMOs) have been observed both experimentally and numerically in various prototypical systems in the natural sciences. In the present article, we propose a mathematical model problem which, though analytically simple, exhibits a wide variety of MMO patterns upon variation of a control parameter. One characteristic feature of our model is the presence of three distinct time-scales, provided a singular perturbation parameter is sufficiently small. Using geometric singular perturbation theory and geometric desingularization, we show that the emergence of MMOs in this context is caused by an underlying canard phenomenon. We derive asymptotic formulae for the return map induced by the corresponding flow, which allows us to obtain precise results on the bifurcation (Farey) sequences of the resulting MMO periodic orbits. We prove that the structure of these sequences is determined by the presence of secondary canards. Finally, we perform numerical simulations that show good quantitative agreement with the asymptotics in the relevant parameter regime.