Cubic Perturbations of Symmetric elliptic Hamiltonians of degree four in a Complex domain

We consider arbitrary one-parameter cubic deformations of the Duffing oscillator $x\mathrm{"}=x-x^3$. In the case when the first Melnikov function $M_1$ vanishes, but $M_2\mathrm{\ne }0$ we compute the general form of $M_2$ and study its zeros in a suitable complex domain.