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.


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updated 29 Dec 2013