Windows of periodicity 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).

You can test below that in the vicinity of c3 = -1.75488 the map fc o3 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.

Set a new c value (and press Enter) to see an animation or set N = 1 to watch iterations of fc .

The width of a window. "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 = λ2λ3 ...λn is the product of the map slopes, λj = 2xj in (n-1) noncentral intervals and β = 1 + λ2-1 + (λ2λ3)-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.

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.


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updated 22 Sept 2002