ELEMENTARY GRAPHICS OF CYCLICITY-1 AND CYCLICITY-2

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