The dud canard: Existence of strong canard cycles in $\mathbb R^3$

TL;DR

This paper rigorously describes the emergence of strong canard cycles in three-dimensional slow-fast systems, extending the concept of canard explosion from two to three dimensions and introducing the phenomenon of the 'dud canard' due to non-explosive periodic families.

Contribution

It extends the theory of canard explosions to three-dimensional systems, proving the existence of strong canard cycles in $\

Findings

Existence of a family of periodic orbits following the strong canard.

Extension of canard explosion phenomena from 2D to 3D systems.

Identification of the 'dud canard' phenomenon where periodic orbits are not explosive.

Abstract

In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in $\mathbb R^3$ through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove -- in the analytic case only -- that for all $0<\epsilon\ll 1$ there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to $\mathcal O(1)$ cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in $\mathbb R^2$, but in contrast to the planar case, the family of periodic orbits in $\mathbb R^3$ is not explosive. For this reason, we have chosen to call the phenomena in $\mathbb R^3$, the ``dud canard''. The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in $\mathbb R^2$, we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit $\epsilon=0$. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic.