Perturbations of symmetric elliptic Hamiltonians of degree four in a   complex domain
  The cyclicity of the exterior period annulus of the asymmetrically perturbed Duffing oscillator is a well known problem extensively studied in the literature. In the present paper we provide a complete bifurcation diagram for the number of the zeros of the associated Melnikov function in a suitable complex domain. 
Bassem Ben Hamed, Ameni Gargouri and Lubomir Gavrilov
math.DS