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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.
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