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.