The paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions. We follow a geometric approach based on the use of C k -normal forms, centre manifolds and (family) blow up, as we did in (Trans. Amer. Math. Soc., to appear). In (Trans. Amer. Math. Soc., to appear) we considered the existence of manifolds of canard solutions for given appropriate boundary conditions. These manifolds need not be smooth at the turning point. In this paper we essentially study the transition time along such manifolds, as well as the divergence integral, providing a structure theorem for these integrals. As a consequence we get a nice structure theorem for the transition equation, governing the canard solutions. It permits to compare different control manifolds and to obtain a precise description of the entry-exit relation of different canard solutions. Attention is also given to the special case in which the canard manifolds are smooth, i.e. when "formal" canard solutions exist.

We study the cyclicity of limit periodic sets that occur in families of vector fields of slow–fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve. All existing results deal with a non-zero slow movement, permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular, our study includes the cyclicity of the slow–fast two-saddle cycle, formed by a regular saddle connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow-divergence integral.

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We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov–Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov–Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

The paper deals with planar slow-fast cycles containing a unique generic turning point. We address the question on how to study canard cycles when the slow dynamics can be singular at the turning point. We more precisely accept a generic saddle-node bifurcation to pass through the turning point. It reveals that in this case the slow divergence integral is no longer the good tool to use, but its derivative with respect to the layer variable still is. We provide general results as well as a number of applications. We show how to treat the open problems presented in Artés et al. (2009) [1] and Dumortier and Rousseau (2009) [13], dealing respectively with the graphics DI 2a and DF 1a from Dumortier et al. (1994) [14].