Elementary graphics of cyclicity 1 and 2

Abstract

In this paper we elaborate the techniques to prove for several elementary graphics that their cyclicity is one or two. We first prove two main results for C^infinity vector fields in general. The first one states that a graphic through an arbitrary number of attracting hyperbolic saddles (hyperbolicity ratio r>1) and attracting semi-hyperbolic points (one negative eigenvalue) has cyclicity 1. A second result says that for a graphic with one hyperbolic and one semi-hyperbolic singularity of opposite character the cyclicity is two. We then specialize to graphics with fixed connections and show that 33 graphics appearing among quadratic systems and listed in a previous paper have a cyclicity at most two (five cases are done only under generic conditions).