Lecture 3 - Thu 2024 09 12
Continue with flows on the line and start with bifurcations.
% ============================================================================= %
                         Left for students to read from prior lecture summaries:
From Lecture 01; the last point in the summary:
                 EXTRA STUFF: Attractors, Tipping points/Bifurcations, etc.

% ============================================================================= %
                                        POINTS LEFT HANGING FROM PRIOR LECTURES:
1) Geometrical view and failure of linearization for 1-D \dot{x} = f(x).
   This happens when df/dx = 0 at a critical point f(x_c) = 0, and the solution
   is dominanted by the first nonlinear term in the Taylor expansion near x_c.
Geometrically it is then easy to see what happens: Draw all the possible cases.
Four, in fact, corresponding to
the pictures given by    f(x) = \pm (x-x_c)^2n and  f(x) = \pm (x-x_c)^{2n+1}
%
% ----------------------------------------------------------------------------- %
2) EXISTENCE AND UNIQUENESS THEOREM for ode. Lipschitz.
2.1) Example of non-uniqueness: Leaky bucket.
     Math.:    \dot{x} = -sqrt{|x|}; x >= 0 [x = depth of water].
     Maybe example where "well posed" fails: why "thermal CSI" does not work.

2.2) Where is chaos and the butterfly?
     Theorem says that, for \dot{Y} = F(Y), Y(0) = Y_0, and F nice enough, the
     solution depends continuously on Y_0. So, small changes in Y_0 cause small
changes in Y(t). Hence: Where is chaos here? Does this preclude it?
No!
Precise statement: if Y1(t) and Y2(t) solve \dot{Y} = F(Y) with initial data
  that are close enough,
  then                                 |Y1(t) - Y2(t)| <= |Y10-Y20| exp(K*t),
  where K is the "Lipschitz" constant
  for F. Also, note that this is for a finite time, because the theorem does
  not guarantee existence for all times.

Now: This is a "universal" bound, which for "well behaved" solutions is a GROSS
over-estimate.  Chaos can happen when it is not an over-estimate, and the error
grows exponentially (i.e.: like e^{mu*t}, for some mu > 0),  which does not the
theorem contradict -- Note: exponential growth of the error is a very important
requirement for chaos ... but not the only one.
%
% ----------------------------------------------------------------------------- %
3) Cover the metatheorem (end of lecture 2)
%
% ----------------------------------------------------------------------------- %
4) The wire and goo system:                            \dot{x} = f(x) = - V'(x)
   Interpret critical points as max/min of V and stability.

   About neglecting inertia:  What does it mean that inertia is small?
   The importance of Non-dimensional variables:

Example [non-dim a damped mass-spring system]:  m \ddot{x} + b \dot{x} + k x = 0.
==> Use as time unit T = b/k [balance of spring with
    dissipation] then                         epsilon*\ddot{x} + \dot{x} + x = 0,
    where epsilon = m*k/b^2.
    Small inertia means epsilon is small.
Also: see the notes "Overdamped approximation. Justification." in this web page.

It is not always OK to neglect terms multiplied by a small parameter; but before
you can talk about "small" or "large" you need to have the equation written in
a-dimensional variables. It does not make sense to compare terms with different
units [e.g.: "m smaller than b" is non-sense!]
Note however, that when you select time scales (and other scales) to write the
equations in a-dimensional form, there is the implication that those scales are
relevant for the solutions that you are interested in!
     Neglecting terms (even when allowed) cannot be done  % \
     for "all" solutions; only for certain classes of     %  | *** [A]
     solutions that are of interest for whatever reasons. % /
Forgetting this leads to "paradoxes", of which there have been many in the
history of science, until how [A] works was properly understood.   We will
not get into this here.

Note: astronomers and fluid dynamicists have developed very powerful methods
      to compute solutions of dynamical systems which are a small perturbation
of the solutions of a known system. We will see some of them later.

Mention example of Neptune, Pluto, and Charon.

% ============================================================================= %
                                                                      NEW POINT.
Numerical methods: Talk about resolution, tolerance, and "numerical
chaos". Example: "chaotic" planet orbits in 2-body problem!
%
% ----------------------------------------------------------------------------- %