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.

We refine the transformation to smooth normal form for an analytic family of vector fields in the neighbourhood of a saddle-node. This refinement is very powerful and allows us to prove the finite cyclicity of families ofgraphics (‘ensembles’) occurring inside analytic families of vector fields. In [ZR1] and [ZR2] it is used to prove the finite cyclicity of graphics through a nilpotent singular point of elliptic type. Several examples are presented: lips, graphics with two subsequent lips, graphics with a nilpotent point of elliptic type and a saddle-node.We also discuss the bifurcation diagram of limit cycles for a graphic in the lips.

A cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate to

We define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent to

Our main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to