ABELIAN-INTEGRALS IN UNFOLDINGS OF CODIMENSION 3 SINGULAR PLANAR VECTOR-FIELDS

In this work it is shown that, for small beta(i), the system x = y, y = +/- x + alphax3 + xy + beta0 + beta1y + beta2x2y has at most two limit cycles when alpha is-an-element-of (- 1/8, infinity)\{0} (Part II) and also when alpha < -1/8 (Part III). Part I contains an introduction to the problem, applications of Abelian integrals and some general results.