The number of zeros of Abelian integrals for a perturbation of a hyper-elliptic Hamiltonian system with a nilpotent center and a cuspidal loop

摘要

In this paper we consider the number of isolated zeros of Abelian integralsassociated to the perturbed system x˙ = y, y˙ = ?x3(x ? 1)2 + #(a + bx + gx3)y, where# > 0 is small and a, b, g 2 R. The unperturbed system has a cuspidal loop and anilpotent center. It is proved that three is the upper bound for the number of isolatedzeros of Abelian integrals, and there exists some a, b and g such that the Abelianintegrals could have three zeros which means three limit cycles could bifurcate fromthe nilpotent center and period annulus. The proof is based on a Chebyshev criterionfor Abelian integrals, asymptotic behaviors of Abelian integrals and some techniquesfrom polynomial algebra.