BIFURCATION OF BIG PERIODIC ORBITS THROUGH SYMMETRIC HOMOCLINICS, APPLICATION TO DUFFING EQUATION

Abstract

We consider a planar symmetric vector eld that undergoes a homoclinic bifurcation. In order to study the existence of exterior pe- riodic solutions of the vector eld around broken symmetric homoclinic orbits, we investigate the existence of xed 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 Dung equation.