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 [Simpson and Meiss (2009 Nonlinearity 22 1123-44)] 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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机译：由分段平滑连续图的边界碰撞分叉形成的锁模区域（共振舌）通常显示出独特的香肠状几何形状，其夹点称为“收缩点”。在本文中，我们扩展了分段线性情况的展开[Simpson and Meiss（2009 Nonlinearity 22 1123-44）]，以显示收缩点如何被非线性破坏。对于N维图，我们获得了该收缩点分叉的余维3展开。我们表明，收缩点的破坏通常是通过创建使鞍形边界变光滑的鞍形节点分叉曲线而发生的，而在另一个边界上留下了纽结。
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