Numerical investigation of resonant homoclinic flip bifurcations I : the single homoclinic-doubling case

We perform a numerical study of certain resonant homoclinic flip bifurcations. These codimension-three phenomena act as organising centres for codimension-two homoclinic flip and resonant homoclinic bifurcations. In a recent paper by Homburg and Krauskopf unfoldings for several cases of resonant homoclinic flip bifurcations were presented as bifurcation diagrams on a sphere around the central singularity. This study focuses on a detailed investigation of the simplest case of these unfoldings (not involving homoclinic-doubling cascades) in the model by Sandstede, which was constructed to contain inclination flip and orbit flip bifurcations. By choosing a sufficiently small sphere around the codimension-three point in parameter space, the unfolding conjectured in Homburg and Krauskopf can be found. However, for larger spheres interesting extra codimension-three bifurcations occur, leading to a more complicated structure. If the sphere is taken too small then several curves in the bifurcation diagram are too close to each other to be detected numerically. This means that there is an important trade-off between finding bifurcation curves numerically and introducing bifurcations by enlarging the sphere too much.