Cross-Over for the Sine Map and the Driven Damped Pendulum

Abstract

The sine map fε(x) = x + μ - (1 - ε) sin (2πx)/2π is by iteration known to exhibit a devil's staircase, which becomes complete as ε tends to zero. Here, the basic work of Shenker, concerning the scaling relations at the golden mean, is generalized. For periodic irrationals, the covergence of the step sizes, the minimal distances from cycle elements to zero mod 1, and their average values, are treated. Furthermore, the self-similarity of the step structures provides a set of "similarity-dimensions", as well as a set of "sub-fractals", emphasizing the close connection to Cantor's discontinuum. Also, the driven damped pendulum is considered. Surprisingly, some differences occur concerning the scaling exponents. Based on analog computations, scaling functions are found, and the differences from the sine map results are discussed.