Lecture 25 - Thu 2023 12 07  Continue with Fractals and dimension.
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                                                   MATLAB FOR THE LOGISTIC MAP.
In case you missed it: with the Lecture 20 I posted
a bunch of scripts in a zipped archive "MatLab_for_Logistic_Map.zip".
If you have not done it, you should run these scripts. The "README"
file there explains what the scripts show. Recall that the Logistic
map is:         y = f(x) = r*x*(1-x);  0 <= r <= 4 and 0 <= x <= 1.
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                                                             EXAMPLE DIMENSIONS
Lorenz (sigma=10, r=28, b=8/3)  D_correlation = 2.049 \pm 0.096
                                D_Lyapunov    = 2.062
Rossler (a = b = 0.2, c = 5.7)  D_correlation = 1.986 \pm 0.078
                                D_Lyapunov    = 2.013
Logistic map at r_inf           D_correlation = 0.500 \pm 0.005
                                D_box         ~ 0.538
The Lyapunov dimension is based on the idea that the rates of contraction and
expansion determine a dimension. Like the correlation dimension, it is
specially suited for dynamical systems [give a bit of the intuition here].
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                              EMBEDDING/ATTRACTOR RECONSTRUCTION Takens theorem
Describe how knowledge of a single dynamical variable can, in principle (under
"general conditions") be enough to reconstruct the whole attractor.

Given single "generic" (smooth) signal from attractor, b(t), construct a vector
trajectory using a time delay T > 0.
x(t) = (b(t), b(t-T), b(t-2*T), ..., b(t-N*T))
If N is large enough, the resulting set is diffeomorphic to the attractor. In
fact, N <= 2*d+1, where d is the dimension of the attractor.
See examples below.
(a) "generic" means that the signal is coupled to all the components of the
    attractor, and it is difficult to check. The idea is that, with
    "probability 1" a signal will be generic. If you have more than one, you
    can test and see if they give consistent results (see below).

Practical considerations:
(b) How do to select T?
    In principle any T should work. But if T is too large or too small, the
image attractor will be "squished" or deformed in such a way that measuring
anything with it will be very hard. A rule of thumb is to make it a significant
fraction (1/10 to 1/2) of a typical orbital period (rough mean wave-length
of b(t)).

(c) How do we know the value of N?
    Measure the dimension D_N of the embedded object. Once it "saturates" (i.e.:
making N larger does not change D_N), N is large enough and D_N = d.
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                                                                       EXAMPLES
Attractor is a circle, and b(t) = sin(t).
Then  x = (sint(t), sin(t)cosT - cos(t)sinT).
   For T small, x_2 is too close to  x_1, get skinny ellipse along x_1 =  x_2.
   For T ~ pi,  x_2 is too close to -x_1, get skinny ellipse along x_1 = -x_2.
   Best T = pi/2; then x_2 = -cos(t) and get a nice circle.

Alternative: x = (b, \dot{b}) = (sin(t), cos(t)) ... circle.

Attractor is a Torus, T2 (quasiperiodic with 2 periods attractor). Then, say
   b(t) = sin(t) + cos(pi*t).
   You should try, numerically, the trick, with x = (b(t), b(t-T), b(t-2*T)),
       and see that you get a torus.
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                                                                   ALTERNATIVES
Discrete system and single signal {x_n},
     use: (x_n, x_{n-1}, ...., x_{n-N}).

Smooth signal where time derivatives are accessible, use: (b, b', b'', ...).
   This is the scheme we used for the "manta ray" attractor for pulsating
   detonation waves --- see The_Videos_and_the_Manta_ray_attractor.zip
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Here talk either about:
  The Manta ray attractor in a simple model for detonation waves
or
  Area preserving maps; and the Henon Map.
  See also, in Lecture 19 the "Shear map iteration pictures".
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Next/last lecture: continue/finish and do
     Floquet Theory: bouncing drops; ping-pong; upside down pendulum; etc.
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EOF