Numerical unfoldings of codimension-three resonant homoclinic flipbifurcations

Abstract

Resonant homoclinic flip bifurcations arecodimension-three phenomena that act as organizingcentres for codimension-two inclination flip, orbit flipand eigenvalue-resonance bifurcations of homoclinicorbits to a real saddle. In a recent paper by Homburg andKrauskopf unfoldings for several cases of resonanthomoclinic flip bifurcations were proposed asbifurcation diagrams on a sphere around the centralsingularity.This paper presents a comprehensive numerical investigation into theseunfoldings in a specific three-dimensional vector field, which wasconstructed by Sandstede to explicitly contain inclination flip and orbitflip bifurcations. For both orbit and inclination flips, different cases canbe classified according to the eigenvalues of the saddle point. All possiblecases are treated including complicated ones involvinghomoclinic-doubling cascades and chaos. In each case, by choosing asufficiently small sphere around the codimension-three point inparameter space, the conjectured unfoldings are largely confirmed.However, for larger spheres interesting new codimension-threebifurcations occur, leading to a more complicated bifurcation structure.The results suggest an important trade-off between finding bifurcationcurves numerically and introducing new bifurcations by enlarging thesphere too much.