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.

© 1996-9

1999-05-24