Devil's Staircase

A plot of the Winding Number $W$ resulting from Mode Locking as a function of $\Omega$ for the Circle Map with $K = 1$. At each value of $\Omega$, the Winding Number is some Rational Number. The result is a monotonic increasing ``staircase'' for which the simplest Rational Numbers have the largest steps. For $K = 1$, the Measure of quasiperiodic states ($\Omega$ Irrational) on the $\Omega$-axis has become zero, and the measure of Mode-Locked state has become 1. The Dimension of the Devil's staircase $\approx 0.8700 \pm 3.7\times 10^{-4}$.

See also Cantor Function

References

Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, 1983.

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993.

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 132, 1990.