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Motivated, at first, by an interest in nonchaotic strange attractors, I have attempted a complete characterisation of the phenomenon of mode-locking in periodically and quasiperiodically driven oscillators, numerically and via perturbation calculations. In both the periodically driven and the quasiperiodically driven cases, it appeared as though certain regions in parameter space associated with mode-locking had points at which their width was zero. Now, this is slightly surprising if true, for two reasons. First, there is no analogue to this pinching effect in skew product circle maps. Secondly, at pinching points, if they exist, the flow is extremely regular, and more so than one might naively expect for this highly nonlinear family of flows. I have not been able to prove analytically that mode-locked regions pinch to zero width. However, I have used Mathematica's arbitrary precision to assemble extremely strong numerical evidence that they do.
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