Windows of periodicity scaling
It is a commonly observed feature of chaotic dynamical systems [1] that, as a
system parameter is varied, a stable periodn orbit appears (by a
tangent bifurcation) which then undergoes a
perioddoubling cascade to chaos and finally
terminates via a crisis (in which the unstable
periodn orbit created at the original tangent bifurcation collides with the
npiece chaotic attractor). This parameter range between the tangent
bifurcation and the final crisis is called a periodn 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).
You can test below that in the vicinity of c_{3} = 1.75488
the map f_{c}^{ o3} is "quadraticlike" and iterations
of the map repeat bifurcations of the original quadratic map
f_{c} . This sheds light on the discussed similarity of
windows of periodicity.
Set a new c value (and press Enter) to see an animation or
set N = 1 to watch iterations of f_{c} .
The width of a window. "Linear"
approximation
Consider a periodn window. Under iterations the critical orbit consecutively
cycles through n narrow intervals S_{1} > S_{2}
> S_{3} > ... > S_{1} each of width
s_{j} (we choose S_{1} to include the critical
point x = 0).

Following [1,2]
we expand f_{c}^{on}(x) for small x (in the
narrow central interval S_{1}) and c near its value
c_{c} at superstability of periodn attracting orbit.
We see that the s_{j} are small and the map in the intervals
S_{2}, S_{2}, ... S_{n} may be regarded as
approximately linear; the full quadratic map must be retained for
the central interval. One thus obtains
x_{j+n} ~ Λ_{n}
[x_{j}^{2} + β(c  c_{c} )] ,

where Λ_{n} =
λ_{2}λ_{3}
...λ_{n} is the product of the map
slopes, λ_{j} = 2x_{j}
in (n1) noncentral intervals and β =
1 + λ_{2}^{1} +
(λ_{2}λ_{3})^{1} +
... + Λ_{n}^{1} ~ 1
for large Λ_{n} . We take Λ_{n} at
c = c_{c} and treat it as a constant in narrow window.
Introducing X = Λ_{n} x and
C = β Λ_{n}^{2}
(c  c_{c} ) we get quadratic map
X_{j+n} ~ X_{n}^{2} + C
Therefore the window width is
~ (9/4β)Λ_{n}^{2}
while the width of the central interval scales as
Λ_{n}^{1}.
Numbers
For the biggest period3 window
Λ_{3} = 9.29887 and
β = 0.60754. 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
β Λ_{3}^{2} = 52.5334 times.
On the left picture below you see the whole bifurcation diagram of
f_{c} . Similar image to the right is located in the centeral
band of the biggest period3 window and is stretched by
9 times in the horizontal x and by 54 times in the
vertical c directions.
[1] J.A.Yorke, C.Grebogi, E.Ott, and L.TedeschiniLalli
"Scaling Behavior of Windows in Dissipative Dynamical Systems"
Phys.Rev.Lett. 54, 1095 (1985)
[2] 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 22 Sept 2002