Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps

TL;DR

This paper studies a specific type of bifurcation called homoclinic corners in planar piecewise-smooth maps, showing how they unfold and lead to unstable periodic solutions through border-collision bifurcations.

Contribution

It provides a detailed unfolding of homoclinic corners in planar piecewise-smooth maps, linking border-collision bifurcations to homoclinic phenomena.

Findings

Sequence of border-collision bifurcations limits to a homoclinic corner

All nearby periodic solutions are unstable

Homoclinic corners are a codimension-one bifurcation in piecewise-smooth maps

Abstract

The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its stable manifold intersects a non-differentiable point of its unstable manifold (or vice-versa). This is a codimension-one bifurcation analogous to a homoclinic tangency of a smooth map, referred to here as a homoclinic corner. This paper presents an unfolding of generic homoclinic corners for saddle fixed points of planar piecewise-smooth continuous maps. It is shown that a sequence of border-collision bifurcations limits to a homoclinic corner and that all nearby periodic solutions are unstable.