Unfolding of a Quadratic Integrable System with a Homoclinic Loop

In this paper, we make a complete study of the unfolding of a quadratic integrable system with a homoclinic loop. Making a Poincaré transformation and using some new techniques to estimate the number of zeros of Abelian integrals, we obtain the complete bifurcation diagram and all phase portraits of systems corresponding to different regions in the parameter space. In particular, we prove that two is the maximal number of limit cycles bifurcation from the system under quadratic non-conservative perturbations. Received July 16, 1999, Revised March 15, 2001, Accepted May 25, 2001