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We study a class of multi-parameter three-dimensional systems of ordinarydifferential equations that exhibit dynamics on three distinct timescales. Weapply geometric singular perturbation theory to explore the dependence of thegeometry of these systems on their parameters, with a focus on mixed-modeoscillations (MMOs) and their bifurcations. In particular, we uncover a novelgeometric mechanism that encodes the transition from MMOs with single epochs ofsmall-amplitude oscillations (SAOs) to those with double-epoch SAOs. Weidentify a relatively simple prototypical three-timescale system that realisesour mechanism, featuring a one-dimensional $S$-shaped supercritical manifoldthat is embedded into a two-dimensional $S$-shaped critical manifold in asymmetric fashion. We show that the Koper model from chemical kinetics ismerely a particular realisation of that prototypical system for a specificchoice of parameters; in particular, we explain the robust occurrence ofmixed-mode dynamics with double epochs of SAOs therein. Finally, we argue thatour geometric mechanism can elucidate the mixed-mode dynamics of morecomplicated systems with a similar underlying geometry, such as of athree-dimensional, three-timescale reduction of the Hodgkin-Huxley equationsfrom mathematical neuroscience.

Bifurcations of mixed-mode oscillations in three-timescale systems: an extended prototypical example