The dud canard: Existence of strong canard cycles in R³

In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in R³ 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<ϵ≪1 there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to 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 R², but in contrast to the planar case, the family of periodic orbits in R³ is not explosive. For this reason, we have chosen to call the phenomena in R³, 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 R², 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 ϵ=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.