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arXiv · 1603.04932

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

David J. W. Simpson
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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.

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David J. W. Simpson. 2016-03-16. Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps. https://doi.org/10.1063/1.4954876

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