  We study a class of multi-parameter three-dimensional systems of ordinary
differential equations that exhibit dynamics on three distinct timescales. We
apply geometric singular perturbation theory to explore the dependence of the
geometry of these systems on their parameters, with a focus on mixed-mode
oscillations (MMOs) and their bifurcations. In particular, we uncover a novel
geometric mechanism that encodes the transition from MMOs with single epochs of
small-amplitude oscillations (SAOs) to those with double-epoch SAOs. We
identify a relatively simple prototypical three-timescale system that realises
our mechanism, featuring a one-dimensional $S$-shaped supercritical manifold
that is embedded into a two-dimensional $S$-shaped critical manifold in a
symmetric fashion. We show that the Koper model from chemical kinetics is
merely a particular realisation of that prototypical system for a specific
choice of parameters; in particular, we explain the robust occurrence of
mixed-mode dynamics with double epochs of SAOs therein. Finally, we argue that
our geometric mechanism can elucidate the mixed-mode dynamics of more
complicated systems with a similar underlying geometry, such as of a
three-dimensional, three-timescale reduction of the Hodgkin-Huxley equations
from mathematical neuroscience.
