Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator

Abstract

We present a geometric explanation of a basic mechanism generating mixed-mode oscillations in a prototypical simple model of a chemical oscillator. Our approach is based on geometric singular perturbation theory and canard solutions. We explain how the small oscillations are generated near a special point, which is classified as a folded saddle-node for the reduced problem. The canard solution passing through this point separates small oscillations from large relaxation type oscillations. This allows to define a one-dimensional return map in a natural way. This bimodal map is capable of explaining the observed bifurcation sequence convincingly.