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 Cinfinity 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).