Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops

Abstract

In this paper we present a complete study of quadratic 3-parameter unfoldings of some integrable system belonging to the class Q(3)(R), and having two centers and two unbounded heteroclinic loops. We restrict to unfoldings that are transverse to Q(3)(R), 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. (C) 1997 Academic Press.