Abelian Integrals in Non-symmetric Perturbation of Symmetric Hamiltonian Vector Field

AbstractIt is shown that the vector field x = y, y = -(x3 - x - λ) + ϵ y(α + βx + x2), λ, ϵ small, has 0, 1, 2, or 3 limit cycles. This shows that some of Petov′s estimates concerning nonoscilation of elliptic integrals are not correct. The paper also improves the results of Guckenheimer and Dangelmayer