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arXiv · 0809.3510

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

D. J. W. Simpson·J. D. Meiss
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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.

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D. J. W. Simpson, J. D. Meiss. 2008-09-20. Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps. https://doi.org/10.1088/0951-7715%2F22%2F5%2F009

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