Finite cyclicity of elementary graphics surrounding a focus or center in quadratic systems

Abstract

In this paper we prove that several elementary graphics surrounding a focus or center in quadratic systems have finite cyclicity. This paper represents an additional step in the large program to prove the existence of a uniform bound for the number of limit cycles of a quadratic vector field which we can call the finiteness part of Hilbert's 16th problem for quadratic vector fields. It nearly finishes the part of the program concerned with elementary graphics. In \cite{DRR} this problem was reduced to the proof that 121 graphics have finite cyclicity. The graphics considered here are the hemicycles $(H_4^3)$, $(H_5^3)$ and $(H_6^3)$ together with $(I_{14a}^2)$, $(I_{15a}^2)$, $(I_{15b}^2)$ and $(I_{27}^2)$ in the notation of \cite{DRR}