Saddle Invariant Objects and their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

TL;DR

This paper investigates the complex global invariant manifolds near a specific homoclinic flip bifurcation in a three-dimensional vector field, revealing how these structures organize phase space and influence bifurcation behavior.

Contribution

It provides a detailed analysis of global invariant manifolds and their intersections in Sandstede's model near a homoclinic flip bifurcation of case B, including topological and bifurcation structures.

Findings

Identification of heteroclinic orbits between saddle periodic orbits and equilibria.

Discovery of regions with infinitely many heteroclinic orbits.

Characterization of equilibria emergence and escape at infinity.

Abstract

When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or non-orientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case $\textbf{B}$, which is characterized by the fact that the codimension-two point gives rise to an additional homoclinic bifurcation, namely, a two-homoclinic orbit. To explain how the global manifolds organize phase space, we consider Sandstede's three-dimensional vector field model, which features inclination and orbit flip bifurcations. We compute global invariant manifolds and their intersection sets with a suitable sphere, by means of continuation of suitable two-point boundary problems, to understand their role as separatrices of basins of attracting periodic orbits. We show representative images in phase space and on the sphere, such that we can identify topological properties of the manifolds in the different regions of parameter space and at the homoclinic bifurcations involved. We find heteroclinic orbits between saddle periodic orbits and equilibria, which give rise to regions of infinitely many heteroclinic orbits. Additional equilibria exist in Sandstede's model and we compactify phase space to capture how equilibria may emerge from or escape to infinity. We present images of these bifurcation diagrams, where we outline different configurations of equilibria close to homoclinic flip bifurcations of case $\textbf{B}$; furthermore, we characterize the dynamics of Sandstede's model at infinity.