Sector-delayed-Hopf-type mixed-mode oscillations in a prototypical three-time-scale model

Abstract

We consider a three-dimensional three-time-scale system that was first proposed by Krupa et al. (2008) under the additional assumption that two singular perturbation parameters are present in the equations. While the presence of three scales was shown to give rise to canard-induced periodic mixed-mode oscillations (MMOs) (Desroches et al., 2012) in the parameter regime studied by Krupa et al. (2008), we additionally observe mixed-mode patterns that display delayed-Hopf-type behaviour (Neĭshtadt, 1987). We present analytical and numerical evidence for the occurrence of stable periodic dynamics that realises both mechanisms, and we discuss the transition between them. To the best of our knowledge, the resulting mixed sector-delayed-Hopf-type MMO trajectories represent a novel class of mixed-mode dynamics in singularly perturbed systems of ordinary differential equations.