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

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 it...

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Bibliographic Details
Published in:Chaos (Woodbury, N.Y.) N.Y.), 2016-07, Vol.26 (7), p.073105-073105
Main Author: Simpson, D. J. W.
Format: Article
Language:English
Subjects:
Bifurcations
Intersections
Invariants
Manifolds
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Summary: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.
ISSN:1054-1500
1089-7682
DOI:10.1063/1.4954876