With basis in number theory, scaling relations for the sine map have been investigated, generalizing the work of Shenker which concentrated on ‘‘golden’’ rotation numbers (equivalents of the golden mean). Scaling functions are found, and exponents associated with the convergence rate, defined for periodic irrationals. In the light of the fact that the step structure obtained by iterations of the sine map exhibits a kind of self-similarity, correspondences to scaling laws for Cantor’s discontinuum have been studied. We define a ‘‘similarity dimension’’ which is found to scale with Shenker’s exponents, as the sine map attains a zero-slope third-order inflection point. Moreover, we examine the subset of steps having associated a denominator, which is a power of 2, and find a correlated fractal dimension which is half the fractal dimension of the original Cantor set. Inspired by these results, we have considered the driven damped pendulum equation and tried to recover corresponding scaling laws. Surprisingly, another scaling exponent z̃apeq21.5 is found where Shenker’s exponent zapeq21.05 was to be expected. Finally, the period-doubling route to chaos inside the hysteretic region for the pendulum equation is discussed, and qualitative agreement with the behavior of the one-dimensional sine map obtained.

• Received 18 February 1986

DOI:https://doi.org/10.1103/PhysRevA.34.2220