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

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

2010-02-11

arxiv.org/abs/1002.2237v1

Nonlinearity 23: 3091-3118 (2010)

Mathematics

Dynamical Systems

32 pages, 10 figures

Scientific paper

10.1088/0951-7715/23/12/006

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.

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