Mixed-Mode Oscillations and Twin Canard Orbits in an Autocatalytic Chemical Reaction

A mixed-mode oscillation (MMO) is a complex waveform with a pattern of alternating small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs). MMOs have been observed experimentally in many physical and biological applications, but most notably in chemical reactions. We are interested in MMOs of an autocatalytic chemical reaction that can be modeled by a system of three ordinary differential equations with one fast and two slow variables. This difference in time scales provides a mechanism for generating small and large oscillations. Provided the time-scale ratio epsilon is sufficiently small, geometric singular perturbation theory predicts the existence of two-dimensional locally invariant manifolds called slow manifolds. Slow manifolds and their intersections, which occur along so-called canard orbits, give great insight into the mechanisms for generating SAOs. The mechanisms for LAOs are less well understood and involve analysis of the global dynamics. We study the autocatalytic reaction model in a parameter regime with epsilon relatively large and observe very complex behavior. We find that for larger values of epsilon, the structure of the slow manifolds is more intricate than what is predicted by the theory for sufficiently small epsilon. Canard orbits in this parameter regime are organized in pairs that have the same number of SAOs. Our results suggest a mechanism where SAOs transform into LAOs and change the geometry of global returns in MMOs.