Cascades of Global Bifurcations and Chaos near a Homoclinic Flip Bifurcation: A Case Study

We study a specific homoclinic bifurcation called a homoclinic flip bifurcation of case C, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, meaning that it can be found as a bifurcation point on a curve of homoclinic bifurcations in a suitable two-parameter plane. In fact, this is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to a homoclinic flip bifurcation of case C. This is a step forward in the understanding of the generic cases of homoclinic flip bifurcations, which started with the study of the simpler cases A and B. In a three-dimensional vector field due to Sandstede, we compute relevant bifurcation curves in the two-parameter bifurcation diagram near the central codimension-two bifurcation in unprecedented detail. We present representative images of invariant manifolds, computed with a boundary value problem set-up, both in phase space and as intersection sets with a suitable sphere. In this way, we are able to identify infinitely many cascades of homoclinic bifurcations that accumulate on specific codimension-one heteroclinic bifurcations between an equilibrium and various saddle periodic orbits. Our computations confirm what is known from theory but also show the existence of bifurcation phenomena that were not considered before. Specifically, we identify the boundaries of the Smale--horseshoe region in the parameter plane, one of which creates a strange attractor that resembles the Rössler attractor. The computation of a winding number reveals a complicated overall bifurcation structure in the wider parameter plane that involves infinitely many further homoclinic flip bifurcations associated with so-called homoclinic bubbles.