Description of the video posted in the canvas web page.
This was a virtual lecture [Tue 2021 11 09 - 18.353 Lecture 18].
It is 95% videos and pictures, with a video introduction to: the Lorenz and
Rossler attractors, chaos in 3-D continuous systems (beyond the phase plane)
and discrete 2-D systems (area preserving maps).
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                                                                        TOPICS.
1) Chaos wheel movie [Lorenz attractor].
2) Rossler lecture [Matlab scripts illustrating the Rossler attractor].
3) Iterations of the shear map pictures.
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                                                              LORENZ EQUATIONS.
Recap on the Lorenz equations.
--- Simple model for convection: project dynamics on just 3 modes.
--- Approximation is rather drastic/poor in convection. Is there a physical
    system for which the Lorenz equations are a good approximation?
    YES: the Malkus-Howard wheel .................................... video V01
         Early model: coffee cups glued to turn-table.
         Fancier model (later) with many containers to approximate
         "continuum" was much harder to get to work.
         Described in Strogatz book ................................. video V02
    Relate Malkus-Howard "wheel" to Lorenz: wings of the butterfly
    correspond to wheel rotating left or right, with random switches.
--- Malkus anecdote of "chaos fountain" in Paris .................... video V03
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                                                             ROSSLER EQUATIONS.
   x_dot = -y - z;   y_dot =  x + a*y;  z_dot =  b + z*(x-c);

a, b, c constants. We will fix a = b = 0.2 and vary c (c = 5 = chaos regime).

                      Run MatLab script "Lecture20211109"

Rossler attractor:
--- Simpler system to understand chaos [shown in MatLab movies].
    Attractor: one "apparent surface" where trajectories orbit around a center,
    and "randomly" leave for an excursion "outside" to come down elsewhere.
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Key features shown by computations:
--- Cross-section. Very thin. Looks like a curve, but upon blow up [10^8
    factor] some transversal structure starts to arise [folds].
--- The evolution, as theta grows, of thin rectangle in a (r, z)-radial
    half-plane (r > 0), where x = r*cos(theta) and y = r*sin(theta) shows:
       "Horizontal" stretching [by a factor ~2],
       Exponential "vertical" compression, and
       FOLDING.
--- An approximate 1-D Poincar\'e map (after one 2*pi turn) can be used to
    describe what happens:
    The attractor is the result of applying the Poincar\'e map infinitely
    many times.

We will SPLIT UNDERSTANDING of what the attractor looks like, and how it
arises in TWO STEPS.
    (1) Vertical structure, z direction [leads to the study of fractals].
    (2) Longitudinal structure, r direction [leads to the study of maps].
                                               WE START NEXT LECTURE WITH THIS.
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                          XTRA (if time allows) ILLUSTRATION OF QUASI-PERIODIC.
The name "quasi-periodic" arises because quasi-periodic functions have the
property that they "almost" repeat with arbitrary accuracy:           video V04
  There is a sequence of quasi-periods 0 < T_1 < T_2 < ... \to \infty
  such that   max_{t} |f(t+T_n) - f(t)|  vanishes as n \to \infty.
This property defines the class of "almost-periodic" functions, which includes
the quasi-periodic ones.
  Quasi-periodic functions can be represented
  by linear combinations of terms of the form   exp(i*(\sum p_j*omega_j)*t)
  where (omega_1, ...., omega_n) are the base
  frequencies and (p_1, ..., p_n) are integers.
  For almost periodic n can be infinite.
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                     XTRA (if time allows) KAM THEORY and AREA PRESERVING MAPS.
Chaos in Hamiltonian systems is the subject of KAM theory.
Poincare maps in Hamiltonian systems are volume preserving and the simplest
setting where chaos arises is for 2-D area preserving maps.
.................................................... show shear_map_iterations.
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