Resonance near Border-Collision Bifurcations in Piecewise-Smooth, Continuous Maps

Abstract

Mode-locking regions (resonance tongues) formed by border-collision bifurcations of piecewise-smooth, continuous maps commonly exhibit a distinctive sausage-like geometry with pinch points called "shrinking points". In this paper we extend our unfolding of the piecewise-linear case [{\em Nonlinearity}, 22(5):1123-1144, 2009] to show how shrinking points are destroyed by nonlinearity. We obtain a codimension-three unfolding of this shrinking point bifurcation for $N$-dimensional maps. We show that the destruction of the shrinking points generically occurs by the creation of a curve of saddle-node bifurcations that smooth one boundary of the sausage, leaving a kink in the other boundary.

32 pages, 10 figures

References in corpus (2)

• Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps
• Symbolic Codes for Rotational Orbits

Cited by in corpus (5)

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• 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
• Border-collision bifurcations in a driven time-delay system
• A synopsis of the non-invertible, two-dimensional, border-collision normal form with applications to power converters