Unfolding of a Quadratic Integrable System with Two Centers and Two Unbounded Heteroclinic Loops

In this paper we present a complete study of quadratic 3-parameter unfoldings of some integrable system belonging to the classQR3, and having two centers and two unbounded heteroclinic loops. We restrict to unfoldings that are transverse toQR3, obtain a versal bifurcation diagram and all global phase portraits, including the precise number and configuration of the limit cycles. It is proved that 3 is the maximal number of limit cycles surrounding a single focus, and only the (1, 1)-configuration can occur in case of simultaneous nests of limit cycles. Essentially the proof relies on a careful analysis of a related non-conservative Abelian integral.