# Windows of regular dynamics scaling

It is a commonly observed feature of chaotic dynamical systems [1] 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"
approximation

Consider a period-n window (see the picture above). 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 period-n 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} )],

كلم *Λ*_{n} = m_{2}m_{3}
...m_{n} is the product of the map slopes,
*m*_{j} = 2x_{j} in *(n-1)* noncentral
intervals and *β = 1 + m*_{2}^{-1} +
(m_{2}m_{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 = βΛ*_{n}^{2}(c - c_{c} )
we get quadratic map

*X*_{j+n} ~ X_{n}^{2} + C.

Therefore the window width is
*~ 9/4βΛ*_{n}^{-2},
while the width of the central interval scales as
*Λ*_{n}^{-1}.
# Numbers

For the biggest period-*3* window *Λ*_{3} = -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 *βΛ*_{3}^{2} =
52.5 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 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
*c*_{3} = -1.7542 . It is
*βΛ*_{3}^{2} = 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
*Λ*_{3} = -9.30 times.
[1] *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)

[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* 12 July 2006