Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps
Dynamical Systems
2016-08-03 v1
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.
Keywords
Cite
@article{arxiv.1603.04932,
title = {Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps},
author = {David J. W. Simpson},
journal= {arXiv preprint arXiv:1603.04932},
year = {2016}
}