Lecture21 Tue 2024 11 14
Chaos and Dynamical systems in 3D ... reduction to 1d maps [in some cases].
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Review, go over the video posted in the canvas web page [virtual lecture from
2021, with videos --- Tue 2021 11 09 - 18.353 Lecture 18]. The lecture covered
the topics below.       The Lorenz equation (first topic below) was covered on
Lecture20 Thu 2024 11 14, but I showed no videos. Also forgot to mention the
Lorenz map. <======================================================== DO IT TODAY
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                                                               LORENZ EQUATIONS.
\dot{x} = sigma (y-x);      % sigma, r, b, ... positive constants.
\dot{y} = r x - y - x z;    % In Lorenz derivation:  sigma = Prandtl  #
\dot{z} = - b z + x y;      %                        r     = Rayleigh #

Invariance: x --> -x and y --> -y.  Phase portrait invariant under this.

Volume contraction:
--- \dot{Y} = F(Y) vector field contracts or expands volume from the sign
                   of div(F).
In this case:                            div(F) = - sigma -1 - b < 0.

Volume decreases exponentially fast --> all trajectories to a set of zero vol.;
   either a fixed point or limit cycle or "strange attractor".
   NOT ALLOWED: neither Quasi-periodic nor repelling fixed points or orbits.
Fixed points:
Origin: stable node for r < 1 and saddle for r > 1.
        Linearization gives \dot{z} = -b z, so game played on x-y plane.
        Pitchfork bifurcation happens at r = 1.

C^+ and C^- (for r > 1):  x = y = \pm sqrt{b(r-1)}; z = r-1.
    Can show: stable for
          r < r_c = sigma (sigma+b+3)/(sigma-b-1)  [assume sigma > b+1]
          at r = r_c a SUBCRITICAL HOPF happens.

What happens for r > r_c ... say sigma = 10, b = 8/3, r = 28.
  Strange attractor: solutions end up in a set of zero volume, but not
                     quite a surface [a fractal]
                     Note: strange attractor is a set containing many solutions.
  Sensitivity to initial conditions.

  Chaos: bounded a-periodic motion with sensitivity to initial conditions.
  Attractor: Invariant bounded set.
             Attracts an open set of I.C.
             It is minimal.
  It is "strange" if it also exhibits sensitivity to initial conditions.
  This excludes: critical points, limit cycles, and quasi-periodic slns.

  Mention: transient chaos [intermittency]; Lorenz map; Floquet exponents.

Note: The Lorenz system is not always chaotic. In some parameter regimes
      it has a globally attracting limit cycle; in others a critical point
is an attractor; and there are also "mixed" regimes where a strange attractor
and a "well behaved" (limit cycle, CP) attractor co-exist.

Physical realization: Malkus-Howard wheel.
To understand better the structure of strange attractors, we will look at a
simpler system: The Rossler attractor.
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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).

                                                    INTRODUCTION and MOTIVATION.
We begin the analysis of the Rossler attractor by studying the dynamics in the
"almost" 1-D "bottom plate". We can do this by looking at the Poincar\'e map of
a cros-section back to itself. Later we will look at the baklava's details. 
This yields a 1-D map from an interval to itself:

x_{n+1} = f(x_n);   map from R to R or from interval to interval.
                    f smooth (or, at least, continuous).
The Rossler Poincare map, as a function of the parameter c. See video.
                    Describe what it does.
Another way to get the map: look at the sequence of local maximums of one of the
        variables, say: x = x(t),  {x_n}.
        Then plot x_{n+1} versus x_n. This gives an "almost" curve.
        Why? Well, remember that the attractor is very flat, almost a surface.
        Hence, for a short time at least, it will behave as a dynamical system
        in the plane, where two initial conditions determine the trajectory.
        In this case the conditions are x = x_n and \dot{x} = 0.
This "prescription" works for any attractor that is very flat, almost a surface.
        In particular, the Lorenz attractor --> The Lorenz map.
        Show how the map looks then.
NOTE: the idea of a map from x_n to x_{n+1} is due to Lorenz.
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%                                                     SIMPLEST EXAMPLE OF CHAOS.
RECALL THE COMPUTER PROBLEM ASSIGNED IN PROBLEM SET #1.
We will revisit it (LATER) in the context of this example.
Because discrete trajectories can cross, 1-D maps have a much greater dynamical
range than 1D continuous systems.  In fact, they "contain" much of the dynamics
of 3D systems such as Rossler, or Lorenz [3-D with nearly 2-D attractor].
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                                     Transversal structure of Rossler. FRACTALS.
Cantor's work on infinity and cardinality.
--- Define cardinality and countable sets (can be ordered).
--- Show rationals have same cardinal as natural numbers.
--- Proof real numbers not enumerable.
--- hypothesis continuum.
Cantor set C_c
--- Measure zero.
--- Cardinal of real numbers.
--- Fractal: Has structure at all scales [self similar C_c ~ 2 C_c].
--- How do we characterize fractals? Dimension ------------------> NEXT LECTURE

In this stuff we will be following
"Lecture21_2021_scan.pdf" and "Lecture21_whiteboard353.pdf"
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