On Reeb components of nonsingular polynomial differential systems on the real plane

Abstract

Denote by r(n) the maximal number of Reeb components that a nonsingular polynomial differential system of degree n on the real plane can have. It is known that r(0)=r(1)=0, r(2)=r(3)=2 and n−1≤r(n)≤n for n≥4. In this paper we prove that r(n)=n for all n≥4. This completely solves, for nonsingular systems, a problem posed by Chicone and Tian in 1982. We prove our result by explicitly constructing nonsingular Hamiltonian systems of degree n presenting n Reeb components for any n≥4.