Canards and curvature: the ‘smallness of ε ’ in slow–fast dynamics

Abstract

A criterion for the existence of canards in singularly perturbed dynamical systems is presented. Canards are counterintuitive solutions that evolve along both attracting and repelling branches of 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. Here, we give quantitative meaning to the frequently encountered statement that the singular perturbation parameter ε , which represents a ratio between fast and slow time scales, is ‘small enough’ for canards to exist. If limit cycles exist, then the criterion expresses the condition that ε must be small enough for there to exist a set of zero-curvature in the neighbourhood of a repelling slow manifold, where orbits can develop inflection points, and thus form the non-convex cycles observed in a canard explosion. We apply the criterion to examples in two and three dimensions, namely to supercritical and subcritical forms of the van der Pol oscillator, and a prototypical three time-scale system with slow passage through a canard explosion.