Devil's Staircase

steps

456

hover for fractions

The mechanical system known as the kicked rotator yields a mapping named the circle map, given by

θ

n+1

=

θ

n

+Ω-

K

2π

sin(2π

θ

n

)

, with

K=1

. The motion is characterized by its winding number

W(Ω)

that represents an average frequency and is independent of the initial value of

θ

0

. The plot of this function is similar to a staircase whose steps appear at definite, ever-increasing rational values

p/q

(use the hover version to see these fractions). After

q

iterations, the value of

θ

n

differs precisely by

p/q

from

θ

0

and the motion is periodic (mode locking). The points between steps are a Cantor set, where the function is discontinuous. The magnification on the bottom-right shows its self-similar structure.