  In multiple time-scale (singularly perturbed) dynamical systems, canards are
counterintuitive solutions that evolve along both attracting and repelling
invariant manifolds. In two dimensions, canards result in periodic oscillations
whose amplitude and period grow in a highly nonlinear way: they are slowly
varying with respect to a control parameter, except for an exponentially small
range of values where they grow extremely rapidly. This sudden growth, called a
canard explosion, has been encountered in many applications ranging from
chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were
initially studied using nonstandard analysis, and later the same results were
proved by standard techniques such as matched asymptotics, invariant manifold
theory and parameter blow-up. More recently, canard-like behaviour has been
linked to surfaces of discontinuity in piecewise-smooth dynamical systems. This
paper provides a new perspective on the canard phenomenon by showing that the
nonstandard analysis of canard explosions can be recast into the framework of
piecewise-smooth dynamical systems. An exponential coordinate scaling is
applied to a singularly perturbed system of ordinary differential equations.
The scaling acts as a lens that resolves dynamics across all time-scales. The
changes of local curvature that are responsible for canard explosions are then
analysed. Regions where different time-scales dominate are separated by
hypersurfaces, and these are pinched together to obtain a piecewise-smooth
system, in which curvature changes manifest as discontinuity-induced
bifurcations. The method is used to classify canards in arbitrary dimensions,
and to derive the parameter values over which canards form either small cycles
(canards without head) or large cycles (canards with head).
