Germs of bifurcation diagrams and SN-SN families

We study the geometry of the bifurcation diagrams of the families of vector fields in the plane. Countable number of pairwise non-equivalent germs of bifurcation diagrams in the two-parameter families is constructed. Previously, this effect was discovered for three parameters only. Our example is related to so-called saddle node (SN)-SN families: unfoldings of vector fields with one saddle-node singular point and one saddle-node cycle. We prove structural stability of this family. By the way, the tools that may be helpful in the proof of structural stability of other generic two-parameter families are developed. One of these tools is the embedding theorem for saddle-node families depending on the parameter. It is proved at the end of the paper.