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 $X_{\epsilon }$ $$X_{\epsilon }:\{\begin{array}{l}\dot{x}=H_y+\epsilon f(x,y)\\ \dot{y}=-H_x+\epsilon g(x,y),\end{array}H=\frac{1}{2}{y}^2+U(x)$$ which bifurcate from the period annuli of $X_0$ for sufficiently small $\epsilon$. Here $U$ is a univariate polynomial of degree four without symmetry, and $f,g$ are arbitrary cubic polynomials in two variables. We take a period annulus and parameterize the related displacement map $d(h,\epsilon )$ by the Hamiltonian value $h$ and by the small parameter $\epsilon$. Let $M_k(h)$ be the $k$-th coefficient in its expansion with respect to $\epsilon$. We establish the general form of $M_k$ and study its zeroes. We deduce that the period annuli of $X_0$ can produce for sufficiently small $\epsilon$, 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 $X_0$ of higher degrees are also studied.