Cubic perturbations of elliptic Hamiltonian vector fields of degree three

The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian vector field XεXεXε:{x˙=Hy+εf(x,y)y˙=−Hx+εg(x,y),H =12y2 +U(x) which bifurcate from the period annuli of X0X0 for sufficiently small ε. Here U   is a univariate polynomial of degree four without symmetry, and f,gf,g are arbitrary cubic polynomials in two variables.We take a period annulus and parameterize the related displacement map d(h,ε)d(h,ε) by the Hamiltonian value h and by the small parameter ε  . Let Mk(h)Mk(h) be the k-th coefficient in its expansion with respect to ε  . We establish the general form of MkMk and study its zeroes. We deduce that the period annuli of X0X0 can produce for sufficiently small ε  , at most 5, 7 or 8 zeroes in the interior eight-loop case, the saddle-loop case, and the exterior eight-loop case respectively. In the interior eight-loop case the bound is exact, while in the saddle-loop case we provide examples of Hamiltonian fields which produce 6 small-amplitude limit cycles. Polynomial perturbations of X0X0 of higher degrees are also studied.