Lecture 22 - Tue 2020 11 17 - Virtual
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Things said, but not stressed, during the last lecture. ************* Read it.
Lecture starts below this.

About the Lorenz attractor [see Strogatz]
-- The system is volume contracting [excludes quasi-periodic orbits with
   two periods or more],
-- It has a trapping region [ellipsoid], with no stable critical points or
   limit cycles.
-- The solutions must end up in a zero volume region, where they are
   a-periodic [not periodic, but they keep wondering around the same zero
               volume region].
   Recall the attractor shape [or run lorenz in MatLab]. A "butterfly" where
   the orbits circle on each wing, and randomly jump from one wing to the
   other.

Note that the "wings" of the butterfly are very thin.
Numerically almost impossible to tell apart from a surface.
But, if they were surfaces, this would produce a contradiction
    [Poincar\'e Bendixon theorem].
Lorenz noticed this and conjectured a puff pastry/baklava type of structure.
    Not just one surface, but many tightly packed together. A near bullseye,
    as we will see later.
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The lecture starts here: Understanding the Rossler attractor.
ROSSLER SYSTEM:    x_dot = -y - z,
                   y_dot =  x + a*y,
                   z_dot =  b + z*(x-c),
where we (numerically) considered the behavior for a = b = 0.2 and 2 < c < 6.0
which includes a period doubling sequence and a chaotic regime.

The MatLab videos shown during the last lecture, show that the attractor can be
understood via the dynamical system Poincare map, as the orbits go around the z
axis. Specifically:   in cylindrical coordinates (r, theta, z),  we look at the
return map for the 1/2 plane z > 0, theta = pi/4.  And, in particular,  to what
happens within the rectangle 3 <~ r <~ 8.5; 1.6 <~ z <~ 3 -- which includes the
attractor:
  This rectangle is stretched (in the radial direction) by the map (by a factor
of 1.5 to 2) and contracted in the z-direction  (by a factor of magniture order
10^{-10})

We split understanding of what the attractor looks like, and how it arises into
TWO STEPS.
(1) Vertical structure, z direction.
(2) Longitudinal structure, r direction.
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% (1) Vertical structure, z direction. % ------------------------------------- %

Replace Poincare map by an idealized version, with expansion is by a factor 2,
and contraction by a factor of 1/3.  Then look at effect of the Poincar\'e map
on a vertical line crossing the attractor.
Note similarity with Cantor set construction:
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% (1.1) Cantor sets. Fractals. % --------------------------------------------- %

1.1a Describe the Cantor set construction and show that the "length" vanishes.
     However, set far from empty. In fact:
1.1b Show Cantor set bijection with numbers on unit interval. See 1.1c below.
     Describe it as 3-adic numbers in [0 1] which do not use digit 1.
     Map then to 2-adic representation of numbers [and viceversa].
1.1c Rational and irrational numbers.
     Show that rationals and integers have the same cardinal.
     p-adic representations of real numbers in [0, 1].
     Introduce p-adic representation: it represents number as a convergent
        sequence. Any number can be represented in this way.
     Representation unique if certain sequences are excluded [for decimal,
        sequences ending with 999999 ...].
     Show that the rational numbers can be ordered [one-to-one map to natural
        numbers].
     Show that the set of all numbers cannot be ordered, using the p-adic
        representation.
1.1d Koch curve and Koch snowflake. Show arc-length goes to infinity.
     Related topic: space filling curves (Sierpinski, Peano, Osgood curves).
1.1e Sierpinski gasket.

"Define" fractals: approximately self-similar with structure at all scales.
     Note that the examples above are all exactly self-similar.
     But in the Cantor construction we could "randomly" shift the position
     of the subtracted middle third, to get "approximately" self-similar.
Note: This definition is not mathematically rigorous because we have not
     specified what "structure" is. For example: a cube is self-similar at
     all scales, but it is not a fractal because it does not have "structure".
     This is "obvious" here, but what does it mean in general? 
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% (1.2) Dimension. % --------------------------------------------------------- %

Started from the desire to characterize fractals.  Would like some "number" so
that two fractals with the same number are "equal" [topologically equivalent].
DIMENSION is an attempt at doing this [but does not quite do it].    There are
several notions of dimension:

1.2a Geometric or regular dimension (integer). % ----------------------------- %
     Works only for sets that are isomorphic to open sets in R^n. Then the
     dimension is the number of variables needed to parameterize the system.
     Note: if we split such a set into many small parts of size r, the number
     of parts N that result behaves like N = O(1/r^n) as r vanishes.

     Examples: examine the situation with intervals, squares, cubes. For
     "regular objects" each dimension gives an additional factor of 1/r for N.
     Look also at an equilateral triangle.

     This gives rise/motivates to the following generalizations:

1.2b Self-similarity dimension. % -------------------------------------------- %
      For an exactly self-similar fractal, the object consists of N parts,
      each a copy of the object scaled by a factor r.
      Let                        d = - log(N)/log(r); so that N = (1/r)^d.
      Then; definition: d = dimension.

       More generally it could be that N ~ c (1/r)^d, so we take

                          d = - lim_{r \to 0} log(N)/log(r).
 
      Example: Cantor set, N=2 and r = 1/3.   Then  0 < d = log(2)/log(3) < 1.

      Example: Koch curve, N=4 and r = 1/3.   Then  1 < d = log(4)/log(3) < 2.

1.2c Box dimension. % -------------------------------------------------------- %
      Let N be the minimum number of "boxes" of size r needed to cover the
      object [the boxes could be balls].
      Then let  d = - lim_{r \to 0} log(N)/log(r)   [if the limit exists].

      Note then that N = O(1/r^d) as r vanishes.
      The Box dimension reduces to the self-similar dimension for
      self-similar fractals.

      Example: box dimension for a non-self-similar fractal [11.4.2 book].
          Pick a Cantor set [or Sierpinski gasket] where stuff removed
          is random interval/square [not always middle third, say].
      Example: box dimension of both rationals and irrationals is 1.      [1]

      [1] poses conceptual problem; the definitions above are too coarse
          to tell the difference between the rationals and real numbers!
      Hence the Hausdorff dimension was introduced.

1.2d Hausdorff dimension. % -------------------------------------------------- %
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