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

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

2008-09-20

arxiv.org/abs/0809.3510v1

Mathematics

Dynamical Systems

27 pages, 6 figures

Scientific paper

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.

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