Lecture 24 - Tue 2020 12 01 - Virtual
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Sensitive dependence on initial conditions.
   Making chaos Rossler/Lorenz style is like "baking": stretch, compress and
   fold --- stretch and fold is, generally, true for all chaos; but compress
   may be missing [e.g.: KAM].

Stretching leads to IC sensitivity. Nearby points in the almost-2D attractor
initially separate at exponential rate: separation ~ c*exp(lambda*t);  where
lambda is the Floquet exponent. Bounds the horizon of predictability:
   Let eps = error in the initial conditions
       del = desired tolerance on sln.
             error after time T. Then:     eps*exp(lambda*T) <= del   [A]
   Note: since lambda > 0 and T > 0, it
         must be eps < del. In fact, if lambda is large,
         this will require, for any reasonable T, that:   eps << del [B]

   Now [A] is the same as                  T  <= (1/lambda)*log(del/eps).
   Thus, for given del and eps,
   the maximum T allowed is                T_m = (1/lambda)*log(del/eps).
   Now, suppose we want to get a
   larger T_m by lowering the IC
   error. Say, n times bigger. Then    n*T_m = (1/lambda)*log(del/eps_n).

   Comparing the two expressions yields:   (del/eps_n) = (del/eps)^n
   That is:
                    epsilon_n = del*(eps/del)^n

   But eps/del < 1, in fact in many situation it is pretty small (see [B]),
   so this quickly yields impossible to achieve errors.

Exponential sensitivity to errors is the basis of the "Butterfly effect"

 From: "https://fractalfoundation.org/resources/what-is-chaos-theory/"
 The Butterfly Effect: This effect grants the power to cause a hurricane
 in China to a butterfly flapping its wings in New Mexico ...

 This sort of language has caused absurd miss-interpretations in the popular
 press. EXPLAIN.
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Takens' embedding theorem.
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How does one characterize [e.g.: measure the dimension] an attractor for which
one does not have access to the full dynamical system?
Example: the work on the "fractal dimension of music" mentioned in an earlier
         lecture.
Example: the Manta Ray attractor for detonations [a pde, but one has access
         to, say, only the wave velocity D = D(t)].
         Describe physics + simple model and the attractor.

The theorem begins by considering an attractor for a smooth, in finite D,
dynamical system, A, for which we only know some function (observable) of the
solution, f=f(t). Then it says that (generically) we can recover the attractor
by, for example, looking at the in R^{n+1} defined by
                     (f(t), f(t-tau), f(t-2*tau), ..., f(t-n*tau)),  [C]
for some natural number
n and tau > 0. That is: the orbits defined by [C] are diffeomorphic to the
original system. But the theorem does not tell how to determine what tau and
n should be, nor how to tell if f is generic.
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The Lorenz map for the Lorenz attractor using successive local maximums.
--- Note the similarity with the Takens' theorem embedding strategy.
--- This works for attractors whose dimension is very close to two:
    e.g.: the Rossler attractor, and the Manta Ray.
Draw pictures.

Use the fact that, for the Lorenz map for the Lorenz attractor |f'(z)| > 1
for all z to "prove" that there are no stable limit cycles.

For the Rossler attractor point out the similarity with the "cross-section"
Poincare map which we constructed numerically in an earlier lecture. This is
the Takens' theorem embedding strategy at work.

Describe the behavior of the "cross-section" Poincare map as parameter values
change, and use to motivate next topic: study of unimodal 1-D maps. A second
motivation is provided by [D]

As we saw earlier, Hopf is excluded for bifurcations from a stable limit
cycle in dissipative systems (volume loss). Then, generally stability is
lost in only one direction,  a situation can be understood by looking at
a 1-D map [Poincare along unstable direction].                            [D]

We will see that in these situations period doubling is the most likely event.             
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