Windows of periodicity scaling
Windows of periodicity
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 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 fco3) which correspond to
two unstable fixed points and 6 points of the stable and unstable
period-3 orbits of fc .
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 fco3 is "quadratic-like" and iterations
of the map repeat bifurcations of the original quadratic map
fc . This sheds light on the discussed similarity of
windows of periodicity.
fcon map renormalization.
The "linear" approximation
Consider a period-n window. 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 =
...λn is the product of the map
slopes, λj = 2xj
in (n-1) noncentral intervals and
1 + λ2-1 +
(λ2 λ3 )-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 = β Λn2
(c - cc ) we get quadratic map
Xj+n ~ Xn2 + C
Therefore the window width is
while the width of the central interval scales as
This scaling is called fcon map renormalization.
For the biggest period-3 window
Λ3 = -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 Λ32 = 52.5334 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 29 Dec 2013