Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps

TL;DR

This paper analyzes the structure and bifurcations of resonance tongues in piecewise-smooth, continuous maps, focusing on the shrinking point bifurcation where these tongues narrow to zero width, providing a symbolic description and bifurcation analysis.

Contribution

It introduces a symbolic framework for rotational solutions with lens-chain structures and unfolds the codimension-two shrinking point bifurcation in N-dimensional maps.

Findings

Characterization of resonance tongue shapes in piecewise-smooth maps

Identification of bifurcation curves emanating from shrinking points

Description of tongue boundary formation in bifurcation diagrams

Abstract

Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark-Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation diagrams. We give a symbolic description of a class of "rotational" periodic solutions that display lens-chain structures for a general $N$-dimensional map. We then unfold the codimension-two, shrinking point bifurcation, where the tongues have zero width. A number of codimension-one bifurcation curves emanate from shrinking points and we determine those that form tongue boundaries.