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Canard-induced phenomena have been extensively studied in the last threedecades, both from the mathematical and from the application viewpoints.Canards in slow-fast systems with (at least) two slow variables, especiallynear folded-node singularities, give an essential generating mechanism forMixed-Mode oscillations (MMOs) in the framework of smooth multiple timescalesystems. There is a wealth of literature on such slow-fast dynamical systemsand many models displaying canard-induced MMOs, in particular in neuroscience.In parallel, since the late 1990s several papers have shown that the canardphenomenon can be faithfully reproduced with piecewise-linear (PWL) systems intwo dimensions although very few results are available in the three-dimensionalcase. The present paper aims to bridge this gap by analysing canonical PWLsystems that display folded singularities, primary and secondary canards, witha similar control of the maximal winding number as in the smooth case. We alsoshow that the singular phase portraits are compatible in both frameworks.Finally, we show on an example how to construct a (linear) global return andobtain robust PWL MMOs.

Canards, folded nodes and mixed-mode oscillations in piecewise-linear slow-fast systems