Lecture 26 - Tue 2020 12 08 - Virtual
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Continue with unimodal maps [logistic map y = f(x) = r*x(1-x)].
Lyapunov exponent; Universality; Feigenbaum numbers; Universal U-Sequence.

The aim is to cover the rest of chapter 10 in Strogatz; pp. 372 to 394.
However, rather than "cover" it, I will fill in some details [useful to
get intuition and understanding] not quite in the book, and I will leave
it up to you to read the book in detail.
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First we look at the behavior of the 2-nd iterate of the map, f2(x) = f(f(x)),
as r increases: quad_f2.pdf
Note that r=2 corresponds to the case when x=1/2 is a super stable fixed point.
and that r=3 corresponds to the birth of the period 2 solution --- which goes
unstable when |f2'| there goes above 2.
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Next we look at the behavior of f3 = f(f(f(x))).
Note the narrow "spaces" between f3 and y=x for r=3.8 at the almost zeros
that correspond to period 3.   This is what causes the intermittency with
almost period 3 intervals in time.
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Now back to the bifurcation diagram in Strogatz page 364. Note the almost
self-similar nature of this fractal structure. What causes this?
For this we look at figure SelfSimQuad.pdf
Note how, beyond the super-stable r=2 case, the graph has an attracting
region that reproduces the complete behavior of the logistic map as r
grows! This means that for 2 < r < r0 = 3.6785 ... f2 reproduces the behavior
of f for 0 < r < 4.
The existence of this region depends only on the qualitative form of the
graph for the logistic map, hence f4 = f2(f2(x)) then reproduces the behavior
of f2, and so on.
.... This fact is the basis for the "renormalization-group" argument that
the book sketches in pages 386--394.
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Now we look at the "sensitive dependence on initial conditions", based on the
existence of a Liapunov exponent which is positive.
We follow the book here, pp. 373--376.   Note, specially, the figure in p376.
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UNIVERSALITY. The bifurcation diagram for some other function, y = r*g(x),
where g is a "single bump function" is remarkably similar for that of the
logistic map [see example for r*sin(pi*x) in page 378, or the one in p 386].
In particular:
1) The sequence in which the periodic solutions appear [the U sequence] is
   the same   1, 2^n, 6, 6*2^n, 5, 5*2^n, 3, ...
2) The ratios that characterize the period 2 bifurcations have the same
   limit values
   lim delta lim (r_{n}-r_{n-1})/(r_{n+1}-r_{n}) = 4.669...
   and alpha = -2.5029 ...  [see figure in page 380].
   The "Feigenbaum" numbers.
   These only depend on y = r*g(x), where g is strictly concave, smooth,
   with a single quadratic maximum and g(0) = g(1) = 0.
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Experiments: Note that the period doubling cascade and the Feigenbaum number
delta has been observed in several experiments. Note also that only a very
few period doublings are accessible experimentally see table in page 383.
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Note that the stuff in pages 383--385 of the book is a short summary of what
I showed you a couple of weeks ago for the Rossler system, numerically.
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Finally: briefly talk about characterizing chaos by it having "continuum"
         spectrum.
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