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.In this paper we present a numerical method to detect and then follow branches of canard orbits in a system parameter. More specifically, we define well-posed two-point boundary value problems (BVPs) that represent orbit segments on the slow manifolds, and we continue their solution families with the package AUTO. In this way, we are able to deal effectively with the numerical challenge of strong attraction to and strong repulsion from the slow manifolds. Canard orbits are detected as the transverse intersection points of the curves along which attracting and repelling slow manifolds intersect a suitable section (near a folded node). These intersection points correspond to a unique pair of orbits segments, one on the attracting and one on the repelling slow manifold. After concatenation of the respective pairs of orbit segments, all detected canard orbits are represented as solutions of a single BVP, which allows us to continue them in system parameters. We demonstrate with two examples—the self-coupled FitzHugh–Nagumo system and a three-dimensional reduced Hodgkin–Huxley model—that branches of canard orbits can be computed reliably. Furthermore, our computations illustrate that the continuation of canard orbits allows one to find and investigate new types of dynamics, such as the interaction between canard orbits and a saddle periodic orbit that is generated in a singular Hopf bifurcation.