# Windows of periodicity scaling

## Windows of periodicity

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. 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 to the left is similar to the whole
bifurcation diagram (see also at the bottom of the page).
For *c = -1.75* in the period-3 window stable and unstable
period-3 orbits appear by a tangent bifurcation.
The stable period-3 orbit is shown to the left below.
If *N = 3* is set (see to the right), we get 8 intersections
(fixed points of *f*_{c}^{o3}) which correspond to
two unstable fixed points and 6 points of the stable and unstable
period-3 orbits of *f*_{c }.

On the left picture the stable period-3 orbit goes through two
"linear" and one central quadratic regions of the blue
parabola. Therefore in the vicinity of *x = 0*
the map *f*_{c}^{o3} is "quadratic-like" 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.
*f*_{c}^{on} map renormalization.
The "linear" approximation

Consider a period-n 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 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} )] ,

where *Λ*_{n} =
λ_{2 }λ_{3 }
...λ_{n} is the product of the map
slopes, *λ*_{j} = 2x_{j}
in *(n-1)* 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}.
This scaling is called *f*_{c}^{on} map renormalization.
## Numbers

For the biggest period-*3* 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 period-*3* 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.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* 29 Dec 2013