The paper presents a careful numerical study of two homoclinic codimension-three bifurcations. In these resonant homoclinic flip bifurcations both a resonance between real eigenvalues and some codimension-two flip bifurcation (inclination flip or orbit flip) occur simultaneously.
Using a model equation of Sandstede the authors study with AUTO and HOMCONT transitions between different regions. Although they consider a specific equation, many of the results should hold universally.
In each case, a small sphere around the codimension-three point is considered assuming that the qualitative picture of the bifurcation curves on the sphere does not depend on the radius. Some care has to be paid to a “good” choice of the radius since for large radii new bifurcations appear while it is impossible to distinguish certain bifurcation curves when the radius is too small. A rich variety of bifurcations can be detected including homoclinic-doubling cascades, torus bifurcations and shift dynamics.
The numerical results largely confirm recent theoretical studies and conjectures of A. J. Homburg and B. Krauskopf [J. Dyn. Differ. Equ. 12, 807-850 (2000; Zbl 0990.37041)].