Pinching of canards and folded nodes: nonsmooth approximation of slow-fast dynamics

Sudden changes in a dynamical system can be modelled by mixtures of slow and fast
timescales, or by combining smooth change with sudden switching. In sets of ordinary
differential equations, the former are modelled using singular perturbations, the latter using
discontinuities. The relation between the two is not well understood, and here we develop
a method called pinching, which approximates a singularly perturbed dynamical system by
a discontinuous one, by making a discontinuous change of variables. We study pinching in
the context of the canard phenomenon at a folded node. The folded node is a singularity
associated with loss of normal hyperbolicity in slow-fast systems with (at least) two slow
variables, and canards are special solutions that characterize the local dynamics. Pinching
yields an approximation in terms of the two-fold singularity of discontinuous (Filippov)
systems, which arises generically in three or more dimensions, and remains a subject of
interest in its own right.