Numerical continuation of canard orbits in slow–fast dynamical systems

Abstract

A trajectory of a system with two clearly separated time scales generally consists of fast segments (or jumps) followed by slow segments where the trajectory follows an attracting part of a slow manifold. The switch back to fast dynamics typically occurs when the trajectory passes a fold with respect to a fast direction. A special role is played by trajectories known as canard orbits, which do not jump at a fold but, instead, follow a repelling slow manifold for some time. We concentrate here on the case of a slow–fast system with two slow and one fast variable, where canard orbits arise geometrically as intersection curves of two-dimensional attracting and repelling slow manifolds. Canard orbits are intimately related to the dynamics near special points known as folded singularities, which in turn have been shown to explain small-amplitude oscillations that can be found as part of so-called mixed-mode oscillations.