Lecture 24 - Tue 2024 12 03 One dimensional maps (Unimodal; logistic map).
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Define unimodal. Example: y = r*x*(1-x); 0 \leq 0.25*r, x, \leq 1.
Describe how to make the orbit diagram.
Describe what the generic orbit diagram looks like.
  Period doublings at sequence r_n [period 2^(n-1) begins], n > 1.
  Super-stable attractor sequence r_n < s_n < r_{n+1}.
  Geometric convergence to a limit r_\infty:   r_\infty - r_n ~ O(\delta^{-n})
                                               r_\infty - s_n ~ O(\delta^{-n})
   \delta = lim_{n \to \infty} (r_{n-1}-r_{n-2})/(r_{n}-r_{n-1})
          = lim_{n \to \infty} (s_{n-1}-s_{n-2})/(s_{n}-s_{n-1})
   Feigenbaum number.
   Universality.      Feigenbaum constant \delta_2 = 4.6692 . . .
Chaos happens for r \geq r_\infty
Periodic windows within chaos region.
Intermittent chaos.
................... Objective is to explain/understand how all of
                    this comes to be, but first:
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Chaos requires sensitivity to small perturbations.        The Liapunov exponent.
See scan of hand-written lecture notes
    Scan_for_Lecture20_2021.pdf pp. 3--4
Discrete and continuum definition.
Implications for predictability of a Liapunov exponent > 0.     DONE IN PRIOR
    Doubling the time requires squaring the data precision.     LECTURE
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                                                        EXTRAS FOR THIS LECTURE
Scan_for_Lecture20_2021.pdf ....... pp. 1-4 correspond to this lecture.
                                    pp. 5-7 correspond to prior lectures.
Orbit_Diagram_Logistic_Map.zip .... contains various details of the orbit
                                    diagram for the logistic map.
The_Logistic_Map_Iterates.zip ..... pictures for the logistic map iterates.
                                    Relevant for this or next lecture.
MatLab_for_Logistic_Map ........... Includes scripts illustrating some of the
                                    points made in the lectures. You should run
                                    these scripts. The "README" file there
                                    explains what each scripts does.
                                    Recall that the Logistic map is:
                                    y = f(x) = r*x*(1-x);
                                    0 <= r <= 4 and 0 <= x <= 1.
[UNM] Unimodal Maps Notes ......... Summary notes for covered topics.
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%                                              Liapunov exponent related issues.
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Consider a generic 1-D map dynamical system: x_{n+1} = f(x_n). Let {z_n} be a
periodic solution sequence:    z_{n+1} = f(z_n) and z_{n+p} = z_n, some p > 0.
Linearizing near {z_n} [i.e.:
write x_n = z_n + d_n; d_n infinitesimal] yields      d_{n+1} = f'(z_n) d_n [L]
Show that:

1) The Liapunov exponent for {z_n}
   is given by                  Lp = (1/p) \sum_{m \leq n < m+p} log(|f'(z_n)|)
   m arbitrary.
2) The solutions to [L] have the form                            d_n = D_n mu^n
   D_n is periodic (D_{n+p} = D_n) and mu^p = f'(z_1) .... f'(z_p).

Note that (2) is a discrete version of Floquet Theory [which I hope we will see
on Lecture 26]. Why is Floquet important:
    (i) Stability of periodic solutions;
   (ii) Stabilization/de-stabilization by parametric forcing. Recall example of
        upside pendulum in one of the videos in Lecture 20.
        "Inverted pendulum" Science Demonstrations (Harvard) ... YouTube.
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                        Start with analysis of orbital diagram for Logistic Map.
Cover, from UNM: [#2 Done Lecture 23], [#3 See first point in this lecture],
[#5, points 1, 2, 3, 5. See first point in this lecture], [#6].
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