Bifurcation of big periodic orbits through symmetric homoclinics‎, ‎application to Duffing equation

‎We consider a planar symmetric vector field that undergoes a homoclinic bifurcation‎. ‎In order to study the existence of exterior periodic solutions of the vector field around broken symmetric homoclinic orbits‎, ‎we investigate the existence of fixed points of the exterior Poincare map around these orbits‎. This Poincare map is the result of the combination of flows inside and outside the homoclinic orbits. It shows how ‎a big periodic orbit converts to two small periodic orbits by passing through a double homoclinic structure‎. Finally‎, ‎we use the results to investigate the existence of periodic solutions of the perturbed Duffing equation.