Lecture20 Thu 2024 11 14
Begin with Chaos. Dynamical systems in 3D
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Detail left out in prior lecture.   EXAMPLE OF HOMOCLINIC BIF. OF A LIMIT CYCLE.

Start with a conservative system:                       \ddot{x} + V'(x) = 0; [A]
with energy E = 0.5*(\dot{x})^2 + V, which has
a saddle at the origin, a center at, say x=1, and an homoclinic orbit for the
saddle enclosing the center. This is easy to achieve with V a cubic with a
local maximum at x = 0, and a local minimum at x = 1.

Then modify the equation to              \ddot{x} + nu(E)\dot{x} + V'(x) = 0; [B]
and make nu negative inside one of the
level lines of E = E0 corresponding to a periodic orbit around the center of [A],
and positive outside. This makes the center into an unstable spiral, and E = E0
a stable limit cycle. Increase now E0. When E0 takes the value corresponding to
the homoclinic orbit of [A], an homoclinic bifurcation of a limit cycle happens.

This scenario is not unusual. If a Hopf bifurcation happens around a critical
point, and there is a saddle nearby, as the limit cycle produced by the Hopf
get larger it may "collide" with the saddle, creating an Homoclinic Bif. of a
limit cycle.
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                                        DIFFERENCE BETWEEN CHAOS AND STOCHASTIC.
Chaos is deterministic, albeit hard to compute with precision.
   Same initial data --> same answer.
   With a program in the *same* computer: same thing happens.
Stochastic equations are not deterministic.
   Same initial data --> different answer answer.
   With a program in the *same* computer: same thing will not happen
                         if a decent random number generator is used.
   If (small) noise is added to a chaotic system, noise will be amplified.
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                                                                 INTRO TO CHAOS.
Rough description of how chaos may happen [in 3-D or more, orbits contained
within a bounded set with no stable critical points or limit cycles there].
There is no guarantee that this will yield chaos (it will not, if, for
example the orbits approach a torus corresponding to quasi-periodic orbits),
but it is quite likely --- need the extra condition of SENSITIVITY TO I.C.

Explain sensitivity to I.C., the butterfly paradox (what it means and what it
does *not* mean),
and restrictions it imposes on predictability ............................ [C]
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For [C] note it yields the equation:       delta \approx epsilon1 e^{F*T1} [D]
where delta    = desired upper bound on error in prediction after time T1.
      epsilon1 = initial error.
Note then that doubling the prediction time to T2 = 2*T1, while keeping
delta fixed, requires reducing the error to epsilon2 = epsilon1^2/delta.
This squaring of the initial error required very quickly limits how far in
time one can go while keeping the prediction error "acceptable".
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                                                         SHORT HISTORY OF CHAOS
From the Boltzmann equation and the ergodic hypothesis to Lorenz.
a) What is the ergodic hypothesis.
b) The puzzle of "third integral of motion".
c) The "small divisors" problem ... what happens in a nonlinear system that couples
    (weakly) two oscillators with irrationally related frequencies? Trying
    to expand the solutions in terms of the small coupling parameter leads
    to equations for the high order perturbations similar to
    \ddot{x} + x = sin(n*t) + sin(m*w*t), with w irrational.
    Show what happens.
d) The Birkoff expansion (early 19 hundreds) show that, formally, any Hamiltonian
   system can be reduced (near an harmonic critical point), to one that has a
   maximum number of conserved quantities. This "links" (b) and (c), as this
   expansions exhibits small divisors.
e) For 4-D Hamiltonians can sometimes reduce the dynamics, using a Poincare
   map, to a map in the plane  P_{n+1} = M(P_n) which preserves area.
   Describe what happens; computations show invariant curves etc.
   See the pictures in the web page for the shear map [an example of an area
   preserving map].
f) (a-e) lead to KAM theory [1950's] that begins to unravel the mystery and
   describes what chaos for Hamiltonian systems looks like [at least, low
   dimensional ones].
g) Lorenz equations. Chaos in a 3-D driven-dissipative dynamical system.
   How does it come about as a "toy" model for convection.
   Roughly describe attractor. Connection with fractals [which also have a
   long history, starting somewhere ~1920; e.g.: Julia and Fatou sets].
h) Lorenz was followed by the introduction of even simpler systems with
   chaos: 1-D maps.
i) Finally: of course, in any history of chaos, short or long, one should not
   skip the role of turbulence, where the first experimental observations go
   back to Reynolds ~1880. Note that Lorenz was trying to understand turbulent
   convection.

Moral: in the early XX century it was mostly believed that "complicated"
       behavior was a product of a large number of dimensions [basis for Stat.
Mech.], but the introduction of digital computers in the 1940-1950 decade
showed that this was naive. "Complicated" behavior can arise in as low as 3D
for continuous dynamical systems, or 1D for discrete ones. On the other hand,
a large system does not guarantee "randomness" either ... this is the Fermi-
Pasta-Ulam phenomena, which I will not have time to cover, also discovered
in the late 1940's.

FINAL POINT. Popular accounts (and even relatively high level ones) of chaos
often mention stuff such as "scientists did not know nonlinear systems could
have complicated behavior till the advent of chaos theory in the 1970's". This
is nonsense. They may not have had any good theory for it, but scientists
definitely knew about this since at least the last quarter of the XIX century.
And they definitely paid attention to it. Same thing happens with fractals,
which were not invented in the 1970's. Mathematicians had been playing with
them since ~1920.
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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.

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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).

Next lecture we will talk about this. Make sure to watch the "video lecture",
since this will be the focus of next lecture. I will not repeat the lecture
... what I expect is for you to ask questions about the videos, what was and
what was not clear, etc.
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