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 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).
Following [1,2]
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 =
λ2 λ3
...λn is the product of the map
slopes, λj = 2xj
in (n-1) noncentral intervals and
β =
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
~ (9/4β)Λn-2
while the width of the central interval scales as
Λn-1.
This scaling is called fcon map renormalization.
Numbers
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.
[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.
Contents
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updated 29 Dec 2013