Windows of regular dynamics 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. 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 is similar to the whole bifurcatin diagram (see two pictures at the bottom of the page).

The width of a window. "Linear" approximation

Consider a period-n window (see the picture below). 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 = m2 m3 is the product of the map slopes, mj = 2xj in (n-1) noncentral intervals and β = 1 + m2-1 + (m2 m3 )-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 = βΛn2(c - cc ) we get quadratic map
    Xj+n ~ Xn2 + C.
Therefore the window width is ~ 9/4βΛn-2, while the width of the central interval scales as Λn-1.


For the biggest period-3 window Λ3 = -9.30 and β = 0.607. 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 βΛ32 = 52.5 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 12 July 2006