Lecture 2 - Thu 2020 09 03 - Virtual
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Finish Lecture 1:
Brief history; from Newton to the present ... Complete this section.
Back to definition of dynamical systems ..... Complete this section.
Examples 1-D systems ........................ Do.
EXISTENCE AND UNIQUENESS THEOREM ............ Do.

Continue with 1-D systems.

--- \dot{x} = sin(x). Exact solution: vs phase portrait.
--- \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}.
--- 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.
--- 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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