Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

TL;DR

This paper develops a method to compute connecting orbits to infinity in a quadratic vector field with a complex homoclinic flip bifurcation, revealing how secondary bifurcations accumulate on heteroclinic bifurcations involving infinity.

Contribution

It adapts Lin's method with compactification and blow-up techniques to compute and analyze connecting orbits to infinity in a three-dimensional bifurcation setting.

Findings

Curves of secondary homoclinic bifurcations accumulate on heteroclinic bifurcations involving infinity.

The adapted Lin's method effectively computes connecting orbits to infinity.

The approach enables tracing bifurcation curves in parameter space.

Abstract

We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $\mathbb{R}^3$ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $n$-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of $\mathbb{R}^3$ with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.