Lecture 3 - Thu 2023 09 14
Continue with flows on the line and start with bifurcations.
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Finish with EXISTENCE AND UNIQUENESS.
1) Example of non-uniqueness.
   Leaky bucket.
   Math.:    \dot{x} = -x^{1/3}; x >= 0 [x = depth of water].

   Maybe example where "well posed" fails: why "thermal CSI" does not work.

2) 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! The precise statement is that if  \dot{Y1} = F(Y1), Y1(0) =Y10
                                    and  \dot{Y2} = F(Y2), Y2(0) =Y10
   then
         |Y1(t) - Y2(t)| <= |Y10-Y20| exp(K*t),

   where K is the "Lipschitz" constant for F.

   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 like e^{mu*t}, for some mu > 0 [this is just one,
   very important, requirement for chaos ... but not the only one].
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                                                        Examples of 1-D systems
--- \dot{x} = f(x) = - V'(x) and motivation [wire and goo].
    Interpret critical points as max/min of V and stability.

    About the "goo" and neglecting inertia.
    When is this OK?
    Example [damped mass-spring system]:  m \ddot{x} + b \dot{x} + k x = 0.
    Non-dim:
    ==> 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.  Neglect inertia if epsilon is small.

    We will justify this later. It is not always OK to neglect terms multiplied
    by a small parameter.

    Note: astronomers and fluid dynamicists have developed very powerful
    methods for computing solutions of dynamical systems which are small
    perturbations of a known system. We will see some of them later.
    Mention example of Neptune, Pluto, and Charon.

--- Example of population growth/logistic equation.
    \dot{N} = k*N  with  k = r*(1-N/N_0).
    N_0 = carrying capacity of environment.

                                           % === FROM HERE DOWN TO NEXT LECTURE %
--- Numerical methods.
    Talk about resolution, tolerance, and "numerical chaos".
    Example: "chaotic" planet orbits in 2-body problem!
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                                                                  BIFURCATIONS:
--- Example: tape buckling [column buckling, other bucklings, \
    clickers]. Other examples: lasers, turbulence, etc.        | Done in prior
--- Define it mathematically. Qualitative change in phase      | lectures.
    portrait. Cannot map one into the other.                   /

    We start with 1-D; and CLASSIFY NOW BY DEGREE OF GENERALITY.

--- 1-D \dot{x} = f(x, r). Picture of f "flapping" as r moves.
    Most generic behavior:
    Creation, annihilation of zeros <--- Saddle node bifurcation
    
--- SADDLE NODE BIFURCATION
    [also known as turning point, fold, blue sky, ...]
    Canonical or NORMAL: \dot{x} = r + x^2.
    Bifurcation diagram.
    Critical slow down phenomena.
    Point out "only one case".
    [sign changes in r and x^2, as well as size changes in
     x^2, can be transformed out].

In what follows we will study "all" possible bifurcations, starting from the
most generic, and adding restrictions where-by others can happen. In each
case we will do this first "graphically" [flapping f] and then via the NORMAL
FORM. We will also state when the bifurcations happen, in terms of the
partial derivatives of f(x, r) at the critical point where the bifurcation
happens [the bifurcation point (x_c, r_c)].

"Normal forms" can be justified using the Inverse and implicit theorems.
    The mathematical statement is: there exist a transformation (local, valid
    only near the bifurcation point) reducing the ode to the normal form.
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