The Geometry of Slow Manifolds near a Folded Node

Abstract

This paper is concerned with the geometry of slow manifolds of a dynamical system with one fast and two slow variables. Specifically, we study the dynamics near a folded-node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of two-dimensional attracting and repelling slow manifolds, and they are a key element of slow-fast dynamics. For example, canard solutions are associated with mixed-mode oscillations, where they organize regions with different numbers of small oscillations. We perform a numerical study of the geometry of two-dimensional slow manifolds in the normal form of a folded node in $\mathbb{R}^3$. Namely, we view the part of a slow manifold that is of interest as a one-parameter family of orbit segments up to a suitable cross-section. Hence, it is the solution of a two-point boundary value problem, which we solve by numerical continuation with the package AUTO. The computed family of orbit segments is used to obtain a mesh representation of the manifold as a surface. With this approach we show how the attracting and repelling slow manifolds change in dependence on the eigenvalue ratio $\mu$ associated with the folded-node singularity. At $\mu = 1$ two primary canards bifurcate and secondary canards are created at odd integer values of $\mu$. We compute 24 secondary canards to investigate how they spiral more and more around one of the primary canards. The first sixteen secondary canards are continued in $\mu$ to obtain a numerical bifurcation diagram.