On the number of limit cycles in perturbations of a two dimensional nonlinear differential system

Summary For the nonlinear system , which has a family { _h } of closed orbits, we consider perturbations of the type , whereP andQ are arbitrary polynomials. The abelian integralsA(h) corresponding to this family { _h } are investigated. By deriving differential equations forA(h) and proving monotonicity for quotients of abelian integrals, we obtain results on the number of zeros of abelian integrals and, hence, on the number of closed orbits _h which persist as limit cycles of the perturbed system (^*). In particular, a uniqueness theorem for limit cycles of (^*) with quadratic polynomialsP, Q is proved. Moreover, whenP, Q are of arbitrary degree, a lower bound for the possible number of limit cycles of (^*) is derived.