Cubic Perturbations of Symmetric elliptic Hamiltonians of degree four in   a Complex domain
  We consider arbitrary one-parameter cubic deformations of the Duffing oscillator $x"=x-x^3$. In the case when the first Melnikov function $M_1$ vanishes, but $M_2\neq 0$ we compute the general form of $M_2$ and study its zeros in a suitable complex domain. 
Bassem Ben Hamed, Ameni Gargouri, Lubomir Gavrilov
math.DS