Lecture 19 - Thu 2023 11 14
Finish bifurcations; flows on the Torus; start with chaos. Lorenz Equations.
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From Lecture 18, cover topics listed under "Things I must not forget to say".
In connection with Landau hypothesis: explain the notion of quasi-periodic and
   almost periodic. Flows on the Torus and higher dimension Tori.
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                                                       Chaos: 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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