Unfolding homoclinic connections formed by corner intersections 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.

References in corpus (2)

• Sequences of Periodic Solutions and Infinitely Many Coexisting Attractors in the Border-Collision Normal Form
• Scaling laws for large numbers of coexisting attracting periodic solutions in the border-collision normal form

Cited by in corpus (5)

• Robust Devaney chaos in the two-dimensional border-collision normal form
• Unfolding codimension-two subsumed homoclinic connections in two-dimensional piecewise-linear maps
• Subsumed homoclinic connections and infinitely many coexisting attractors in piecewise-linear maps
• Renormalisation of the two-dimensional border-collision normal form
• Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form