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

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.