Lecture 26 - Tue 2023 12 12
It will be either about (1) or (2) below.
(1) Area preserving maps; the third integral of motion; KAM; the Henon map.
    Extras; see:
    "Lecture 19:  Shear map iteration pictures" and "Henon map below".
(2) Faraday waves; upside down pendulum; Floquet theory; period doubling
    for ping-pong balls; and walking/bouncing droplets.
    Extras; see:
    The video of the 2021/11/09 Lecture; where I showed an inverted pendulum
    [or google Harvard Science Demonstrations]. The video can be found in
    YouTube.
    Also the posted pdf "Bouncing drops and dynamical systems"; which is
    from a lecture by Nicholas Liu [18.353, Lecture 23, 2021 12 02].

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                                                                      HENON MAP
[See Lectures_23-24 pre, from 2020 by Matt Durey, in web page]
Mimics what Rossler does:  x_{n+1} = y_n + 1 - a*x_n^2;
                           y_{n+1} = b*x_n ............. |b| < 1;
Good choice is a = 1.4 and b = 0.3.

Map: Start with a rectangle lined up along the x axis.
     #1  Stretch and fold:        x2 = x1    and  y2 = 1 + y1 - a*x1^2
     #2  Compress along x-axis:   x3 = b*x1  and  y3 = y2
     #3  Flip:                    x4 = y3    and  y4 = x3
Note:
--- Map is invertible, unlike 1-D maps, and very much like the Poincare maps
    for Rossler or Lorenz.
--- For |b| < 1, map contracts. Area goes down by factor |b|.
--- If a is not too large, and b is small enough, map has a trapping region
    in the plane, very much like Rossler or Lorenz.
--- Map does to a rectangle along x-axis something very similar to what Rossler
    does, but by picking b not too small, it is easier to see how the attractor
    is build up by iterating the map.
--- In this map a plays the role of r in the logistic map [or c in Rossler].
    As a grows a period doubling cascade happens, leading to a strange attractor.

Note: for b = 1 the map is area preserving. In fact it was initially proposed this
      way, to mimic what happens to the Poincare Map in a Hamiltonian systems.
      In this case the chaos that occurs is governed by KAM theory.

I encourage you to play with this map on the computer. It is very easy to do,
and you can reproduce many of the phenomena that we have studied in this course.
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