Complete Devil's Staircase, Fractal Dimension, and Universality of Mode- Locking Structure in the Circle Map

Abstract

It is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos. The complementary set to the stability intervals is a Cantor set of fractal dimension $D=0.87$. This exponent is found to be universal for a large class of functions.