Subharmonic bifurcation in the sine map: An infinite hierarchy of cusp bistabilities

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Abstract
The structure of the phase-locked zones in the sine map is examined. Attention is drawn to general features of the map's phase diagram and two, dynamically distinct kinds of bistability are distinguished: One type arises locally through a cusp catastrophe and the other, nonlocally, through crossing (in the parameter plane) of remote stable manifolds. Numerical work indicates that a Cantor set of cusp bistabilities and other related features forms an infinite binary tree in every Arnol'd tongue. (An infinite set of such structures lies arbitrarily close to the parameter line for the quasiperiodic transition to chaos, and also appears in other phase-locked zones.) The binary tree of features obeys the vector scaling found in the quartic map and seems to be generic for multiple-extrema, one-dimensional maps.