Lecture 2 - Tue 2023 09 12
Continue with flows on the line.
% ===============================================================================
Finish "NEXT LECT." points in Lecture 1:
%
Complexity diagram for continuous systems.
Back to dynamical systems ................... Complete this section.
Flows on the line ........................... Do this section.
   Here, left for next lecture (where I will do examples) are:
   --- \dot{x} = f(x) = - V'(x) and motivation [wire and goo].
   --- Population grow, logistic equation.
EXISTENCE AND UNIQUENESS THEOREM ............ Do this section.
   For next lecture:
   1) Example of non-uniqueness.
   2) Where is chaos here? Extreme example, when bound for the
      continuity constant, e^{K*T}, actually happens.
% ----------------------------------------------------------------------------- %
New topic(s) below
% ----------------------------------------------------------------------------- %
Critical points and linearization:
   For a dynamical system, \dot{y} = F(y), a critical (or stationary) point
   is one such that F(x_c) = 0.  Then y = x_c is a solution.
   Many physical systems have equilibrium (time independent) solutions, so
   that critical points are rather important.
   [Note that, for discrete dynamical systems, x_{n+1} = F(x_n), a fixed
    point is defined by F(x_c) = x_c].

   Linearized stability of a critical point:  write y = x_c + z, where z is
   very small ["infinitesimal"],
   then the equations reduce to:              \dot{z} = A z,   [A]
   where A is the Jacobian [matrix of
   partial derivatives] of F at x_c].
   [For a discrete system the linearization yields: z_{n+1} = A z_n,
    and then stability corresponds to abs(eigenvalues) < 1].

   Then x_c is (linearly) stable if all the eigenvalues of A have negative
   real part, and unstable if at least one has a positive real part.
   QUESTION: when does the behavior of [A] accurately reflect the behavior
             of the full system near x_c?
   ANSWER: see the next topic.
   Note: we will see more about subtle details of stability, etc., later in
         the class.

   NOTE: here point out how the "geometric view" of 1-D systems is consistent
         with the linearization approach above, but also allows you to see
         what happens when the "linearize approach" fails. That is, when
         f(x_c) = 0 AND df/dx(x_c) = 0 [f scalar function].
         Draw all the possible cases of this [basically four, corresponding
         to the pictures given
         by            f(x) = \pm (x-x_c)^2n and  f(x) = \pm (x-x_c)^{2n+1}

% ----------------------------------------------------------------------------- %
    The "meta-theorem" [note that this is NOT an actual mathematical theorem,
    but a generic observation of something that is true in many examples].
Statement:
    Suppose that you have a problem where it is possible to linearize to
    answer the question in the "infinitesimal amplitude limit".  Suppose
    further that the linear problem is robust; that is: if you perturb the
    linear system, then the answer/property of the linear system does not
    change.
    Then the answer/property is also true for non-infinitesimal perturbations
    that are "small enough".

    Note: this is related to the concept of  "structural stability", which we
          will introduce in a few lectures.

Example #1:
    Consider several examples of properties of the linear system in [A] which
    are robust, such as (i) All the eigenvalues have negative real part;
    (ii) One eigenvalue has positive real part; (iii) All the eigenvalues are
    distinct and nonzero; etc.
    Properties that rely on these facts carry over to the full system.
    On the other hand, for example (a) An eigenvalue has zero real part, or
    (b) An eigenvalue is real and double; etc are not robust, and properties
    that depend on this may not carry over to the full system.

Example #2: Implicit and Inverse Function theorems.
    State them, and show how they are examples of the "meta-theorem"

    You will not find these theorems in Strogatz book, nor in any "modern"
    calculus book [no longer taught in 18.02], but you can find them in
    OLD calculus books; or in an analysis book.

% ----------------------------------------------------------------------------- %
Final point:
I will NOT prove many things in this course. However, most of the proofs follow
within the context of the "meta-theorem". In particular, for bifurcations, many
proofs "reduce" to cleverly pose the problem so it can be answered using the
implicit function theorem.
% ===============================================================================
EOF