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Notes to Lecture 25 - Thu 2024 12 05
The Logistic Map y = f(x) = r*x*(1-x);  0 <= r <= 4 and 0 <= x <= 1.
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                                                                RENORMALIZATION
We work with the shifted map F of "logimap_lecture(7)" to "explain" why the
results in UNIVERSALITY below hold. Note that what we observe for the logistic
map will be reproduced by any unimodal map [defined below in UNIVERSALITY].
    Note: the script logimap_lecture was posted with Lecture24-2024
          within MatLab_for_Logistic_Map.zip

Consider the successive "green boxes" that happen to the iterates F_q, q=2^n,
at r = s_n. We can now scale all these boxes so that they are the unit square
centered at the origin. In each of them we see a function that "resembles" the
shifted logistic map at r = s_0 = 2. These functions are given by

     g_n(x) = (-1)^n alpha_n F_q(x/alpha_n, s_n)

where alpha_n is the stretching factor and the (-1)^n is to "flip up" all the
squares. We now make the following "renormalization hypothesis" (based on what
we see numerically)
a) s_n ---> r_inf      as  n --> infinity
b) alpha_n ~ alpha^n   for n large,
c) The limit n --> infinity of g_n exists. Call this limit g.

Then it can be seen [apply the limit to g_n and g_{n+1}, both should yield the
same answer] that g must satisfy the universal equation

        g(x) = -alpha g_2(x/alpha)

where g_2 is the second iterate of g. This equation determines both g and
alpha (must be solved numerically). A somewhat similar argument can then be
used to obtain the other Feigenbaum's number, delta.

Note that these arguments are NOT rigorous, and involve hard to proof
hypothesis. In the end, the proof is in the pudding: what they predict
agrees with the numerical observations. These are physics-type arguments,
not rigorous math.
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                                                                   UNIVERSALITY
In the period doubling route to chaos:

1) The 2^n cycle appears for r = r_n, where
   r_1 < r_2 < .... < r_n  ---> r_inf.   Chaos begins past r_inf.
   r_n converges geometrically to r_inf.
   
   Then:  delta = lim (r_n-r_{n-1})/(r_{n+1}-r_n) ~ 4.669...
          is Feigenbaum's number.
   Universality: this is true for any Unimodal map, f_u = r*umap(x)
   where umap is concave with a single max at 0 < x=x_M < 1.
   Then x_M plays the role of 1/2 in the logistic map.

2) The 2^n cycle is super-stable for r = s_n;  r_n < s_n < r_{n+1}.
   Note that x_M is a point in the super-stable cycle.
   Then let x_* be the closest point to x_M in the 2^n cycle, and
   let
       d_n = x_* - x_M.

   Then the d_n alternate signs and converge to zero geometrically,
   with
          lim -d_n/d_{n+1} = alpha = 2.5029 ... also a universal constant.

3) Recall that a 2^n cycle is a fixed point for the f_2^n iterate of the
   map, and that it's stability is given by
      (f_2^n)' = f'(x_1)*f'(x_2)* ... * f'(x_2^n),
   where {x_n} are the points in the cycle.
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