Perturbations from an elliptic Hamiltonian of degree four

Abstract

The paper deals with a complete study of the number of zeros of Abelian integrals, related to perturbations of the Hamilonian vector fields with an elliptic Hamiltonian of degree four: (x) over dot = y, (y) over dot = P(x) + deltaQ(x)y, where P and Q are polynomials of degree respectively 3 and 2, and delta small. We prove that if the unperturbed Hamiltonian vector field has a saddle loop, a cuspidal loop or a global center, then for the perturbed system the corresponding maximum number of zeros of Abelian integral is respectively 2, 4 and 4. In the last case, the perturbed system may have a quadruple limit cycle, as was conjectured in (11).