  Mixed-mode oscillations (MMOs) are complex oscillatory patterns in which
large-amplitude relaxation oscillations (LAOs) alternate with small-amplitude
oscillations (SAOs). MMOs are found in singularly perturbed systems of ordinary
differential equations of slow-fast type, and are typically related to the
presence of so-called folded singularities and the corresponding canard
trajectories in such systems. Here, we introduce a canonical family of
three-dimensional slow-fast systems that exhibit MMOs which are induced by
relaxation-type dynamics, and which are hence based on a "jump mechanism",
rather than on a more standard canard mechanism. In particular, we establish a
correspondence between that family and a class of associated one-dimensional
piecewise affine maps (PAMs) which exhibit MMOs with the same signature.
Finally, we give a preliminary classification of admissible mixed-mode
signatures, and we illustrate our findings with numerical examples.
