Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations

Resonant homoclinic flip bifurcations are codimension-three phenomena that act as organising centres for codimension-two inclination flip, orbit flip and eigenvalue-resonance bifurcations for homoclinic orbits to a real saddle. In a recent paper by Homburg and Krauskopf unfoldings for several cases of resonant homoclinic flip bifurcations were proposed as bifurcation diagrams on a sphere around the central singularity. This paper presents a comprehensive numerical investigation into these unfoldings in a specific three-dimensional vector field, which was constructed by Sandstede to explicitly contain inclination flip and orbit flip bifurcations. For both orbit and inclination flips different cases can be classified according to the eigenvalues of the saddle point. All possible cases are treated including complicated ones involving homoclinic-doubling cascades and chaos. In each case, by choosing a sufficiently small sphere around the codimension-three point in parameter space, the conjectured unfoldings are largely confirmed. However, for larger spheres interesting new codimension-three bifurcations occur, leading to a more complicated bifurcation structure. The results suggest an important trade-off between finding bifurcation curves numerically and introducing new bifurcations by enlarging the sphere too much.