## Devil's Staircase

A plot of the Winding Number resulting from Mode Locking as a function of for the Circle Map with . At each value of , 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 , the Measure of quasiperiodic states ( Irrational) on the -axis has become zero, and the measure of Mode-Locked state has become 1. The Dimension of the Devil's staircase .

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.