A Lin's method approach for detecting all canard orbits arising from a folded node

Abstract

Canard orbits are relevant objects in slow-fast dynamical systems that organize the spiraling of orbits nearby. In three-dimensional vector fields with two slow and one fast variables, canard orbits arise from the intersection between an attracting and a repelling two-dimensional slow manifold. Special points called folded nodes generate such intersections: in a suitable transverse two-dimensional section Σ, the attracting and repelling slow manifolds are counter-rotating spirals that intersect in a finite number of points. We present an implementation of Lin's method that is able to detect all of these intersection points and, hence, all of the canard orbits arising from a folded node. With a boundary-value-problem setup we compute orbit segments on each slow manifold up to Σ, where we require that the corresponding end points in Σ lie in a one-dimensional subspace known as the Lin space Z. The Lin space Z must be transverse to the slow manifolds and it remains fixed during the detection of canard orbits as zeros of the signed distance along Z. During the computation, a tangency of Z with one of the intersection curves in Σ may arise. To overcome this, we update the Lin space at an intermediate continuation step to detect a double tangency of Z to both curves in Σ, after which the canard detection is able to continue. Our method is demonstrated with the examples of the normal form for a folded node and of the Koper model.

Saddle Slow Manifolds and Canard Orbits in R 4 $\mathbb{R}^{4}$ and Application to the Full Hodgkin–Huxley Model

“…In systems of ordinary differential equations with one fast and two slow variables, extended attracting and repelling slow manifolds may intersect transversally in isolated canard orbits. In such systems, canard orbits can be detected by placing a cross-section transverse to both manifolds near the folded node and determining their intersections; see also [ 35 , 45 , 48 , 64 ]. In systems with two fast and two slow variables, attracting and saddle slow manifolds spiral around each other in forward and backward time, respectively, in the vicinity of the folded node.…”