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 (in which the unstable
period-n orbit created at the original tangent bifurcation collides with the
n-piece chaotic attractor). 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 below is similar to the whole
bifurcatin diagram (see two pictures at the bottom of the page).
The width of a window. "Linear"
Consider a period-n window (see the picture above). 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 )],
كلم Λn = m2m3
...mn is the product of the map slopes,
mj = 2xj in (n-1) noncentral
intervals and β = 1 + m2-1 +
... + Λn-1 ~ 1
for large Λn.
We take Λn at c = cc
and treat it as a constant in narrow window.
Introducing X = Λnx 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.
You see below the period-3 Mandelbrot midget placed on complex plane at
c3 = -1.7542 . It is
βΛ32 = 52.5 times lesser
then the whole Mandelbrot set. In a symilar way the J(0) midget
(the black circle in the center of the right picture) is squeezed
Λ3 = -9.30 times.
 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