Homoclinic bifurcations gained a lot of attention because they are closely related to transitions to chaotic dynamics. Many kinds of homoclinic bifurcations were studied (the best known is the Shil’nikov case of a homoclinic orbit to a saddle-focus equilibrium). The paper considers a different case of a homoclinic orbit to a saddle equilibrium of a vector field in with real eigenvalues. The paper is devoted to the study of three-parameter unfoldings of resonant orbit flip and inclination flip homoclinic orbits.
First, all known results on codimension-two unfoldings of homoclinic flip bifurcations are presented. A homoclinic flip bifurcation can be brought about in two ways, called an orbit flip and an inclination flip bifurcation. It is shown that the orbit flip and inclination flip both feature the creation and destruction of a cusp horseshoe. The authors provide a complete overview of the relevant results in the literature.
The main question studied in the paper is what happens when one of the eigenvalue conditions separating the different codimension-two cases is not satisfied. This is called a resonant homoclinic flip bifurcation. It is shown near which resonant flip bifurcations a homoclinic-doubling cascade occurs and the role it plays in the unfoldings of resonant homoclinic flip bifurcations. Three-parameter unfoldings of all possible cases are presented. These three-parameter unfoldings are obtained by glueing to each other corresponding codimension-two unfoldings of homoclinic flip bifurcations on a sphere around the origin (central singularity) in parameter space. Under the assumption that the bifurcation set has cone structure, the bifurcations on the sphere represent the codimension-three unfolding. Although the three-parameter unfoldings obtained in this way are still conjectural in part, they constitute the simplest, consistent glueings that take into account all known information. In particular these unfoldings are proved to contain homoclinic-doubling cascades.