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

TL;DR

This paper investigates how nonlinearity affects the structure of resonance regions near border-collision bifurcations in piecewise-smooth maps, revealing the destruction of shrinking points through saddle-node bifurcations.

Contribution

It extends previous linear analysis to nonlinear cases, providing a codimension-three unfolding and explaining the generic destruction mechanism of shrinking points.

Findings

Shrinking points are destroyed by nonlinearity.

A curve of saddle-node bifurcations smooths one boundary of the resonance sausage.

The analysis applies to N-dimensional maps with a codimension-three unfolding.

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.