Windows of regular dynamics scaling
It is a commonly observed feature of chaotic dynamical systems  that, as a
system parameter is varied, a stable period-n orbit appears (by a
tangent bifurcation) which then undergoes a period-doubling cascade
to chaos and finally terminates via a crisis.
This parameter range between the tangent
bifurcation and the final crisis is called a period-n window.
Note, that the central part of the picture is similar to the whole
bifurcatin diagram (see two pictures at the bottom of the page).
The width of a window. "Linear" approximation
Consider a period-n window (see the picture below). Under iterations the
critical orbit consecutively cycles through n narrow intervals
S1 → S2 → S3 → ...
→ S1 each of width sj
(we choose S1 to include the critical point x = 0).
we expand fcon(x) for small x (in the
narrow central interval S1) and c near its value
cc at superstability of period-n attracting orbit.
We see that the sj are small and the map in the intervals
S2, S2, ... Sn may be regarded as
approximately linear (the full quadratic map must be retained for
the central interval). One thus obtains
xj+n ~ Λn
[xj2 + β(c - cc )],
where Λn = m2 m3
...mn is the product of the map slopes,
mj = 2xj in (n-1) noncentral
intervals and β = 1 + m2-1 +
(m2 m3 )-1 +
... + Λn-1 ~ 1
for large Λn.
We take Λn at c = cc
and treat it as a constant in narrow window.
Introducing X = Λn x and
C = βΛn2(c - cc )
we get quadratic map
Xj+n ~ Xn2 + C.
Therefore the window width is
while the width of the central interval scales as
For the biggest period-3 window Λ3 = -9.30 and
β = 0.607. So the central band is reduced ~ 9 times and
reflected with respect to the x = 0 line as we have seen before.
The width of the window is reduced βΛ32 =
52.5 times. On the left picture below you see the whole bifurcation diagram
of fc. Similar image to the right is located in the centeral
band of the biggest period-3 window and is stretched by
9 times in the horizontal x and by 54 times in the
vertical c directions.
 J.A.Yorke, C.Grebogi, E.Ott, and L.Tedeschini-Lalli
"Scaling Behavior of Windows in Dissipative Dynamical Systems"
Phys.Rev.Lett. 54, 1095 (1985)
 B.R.Hunt, E.Ott
Structure in the
Parameter Dependence of Order and Chaos for the Quadratic Map
J.Phys.A 30 (1997), 7067.
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updated 12 July 2006