Ducks on the torus: existence and uniqueness

TL;DR

This paper demonstrates the existence and uniqueness of canard cycles in generic slow-fast systems on the two-torus with a single parameter, contrasting with the planar case, and analyzes their basin of attraction.

Contribution

It establishes the existence and uniqueness of canard cycles in one-parameter slow-fast systems on the two-torus, a novel result compared to the planar case.

Findings

Existence of canard cycles for small parameter values

Unique attracting and repelling canard cycles coexist

Attracting cycle's basin covers almost the entire torus

Abstract

We show that there exist generic slow-fast systems with only one (time-scaling) parameter on the two-torus, which have canard cycles for arbitrary small values of this parameter. This is in drastic contrast with the planar case, where canards usually occur in two-parametric families. Here we treat systems with a convex slow curve. In this case there is a set of parameter values accumulating to zero for which the system has exactly one attracting and one repelling canard cycle. The basin of the attracting cycle is almost the whole torus.