A Geometric Model for Mixed-Mode Oscillations in a Chemical System

This paper presents a detailed analysis of mixed-mode oscillations in the “autocatalator,” a three-dimensional, two time scale vector field that is a chemical reactor model satisfying the law of mass action. Unlike earlier work, this paper investigates a return map that simultaneously exhibits full rank and rank deficient behavior in different regions of a cross section. Canard trajectories that follow a two-dimensional repelling slow manifold separate these regions. Ultimately, one-dimensional induced maps are constructed from approximations to the return maps. The bifurcations of these induced maps are used to characterize the bifurcations of the mixed-mode oscillations. It is further shown that the mixed-mode oscillations are associated with a singular Hopf bifurcation that occurs in the system.