Lecture 3 - Tue 2020 09 08 - Virtual
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Finish Lecture 2:
Implicit and Inverse Function theorems, and "meta-theorem'.
   Note: please read the "SelectedLectureNotes18385.pdf" file.
         In particular, after today's lecture, see if you can compute
   the "two possible limits" mentioned below equation (1.2.5) there.
Meta-theorem: When the linearized problem is "wimpy", the nonlinear problem
   may behave different from the linear one. When the linearized problem is
   "strong", it usually can be proved that the nonlinear problem behaves in
   the same way.
Formalize wimpy and strong: "structural stability" to linear perturbations.
Examples: Inverse and implicit theorems as well as behavior of a dynamical
          system near a critical point.
Note that this is NOT an actual mathematical theorem!

Continue with 1-D systems.

--- \dot{x} = sin(x). Exact solution:
    t = ln{[(1+cos x0)*sin x]/[(sin x0)*(1+cos x)]}, where x(0) = x0,
    vs phase portrait.
    Easy to check that the derivative of ln{[sin x]/[1+cos x]} is 1/sin x.

--- No oscillations in 1-D.

--- Recap \dot{x} = f(x) = - V'(x) and motivation [wire and goo].
    Interpret critical points as max/min of V and stability.

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

--- Example of non-uniqueness:
    Physical: Empty bucket.
    Math.:    \dot{x} = -x^{1/3}; x >= 0.

--- Back to "goo". When is it OK to neglect inertia?	<<=== Later, with 2-D.
    Non-dim. Example [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.  Neglect intertia if epsilon is small.

--- Numerical methods.					<<=== Next lecture(s)
    Talk about resolution, tolerance, and "numerical chaos".
    Example: "chaotic" planet orbits in 2-body problem!

BIFURCATIONS:
--- Example: tape buckling [column buckling, other bucklings, clickers].
    Other examples: lasers, turbulence, etc.

--- Define it mathematically. Qualitative change in phase
    portrait. Cannot map one into the other.

--- 1-D \dot{x} = f(x, r). Picture of f "flapping" as r moves.
    Bifurcation: creation, annihilation of zeros.
    CLASSIFY NOW BY DEGREE OF GENERALITY.

--- Saddle node bifurcation [turning point, fold, ...]
    Canonical: \dot{x} = r + x^2.
    Bifurcation diagram.
    Critical slow down phenomena.
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