Lecture 12 - Thu 2020 10 08 - Virtual
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Finish index theory.
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Limit cycles. ----------------------------------------
--- Definition. --------------------------------------
--- Stability:  --------------------------------------
    2-D stable, unstable, semi-stable.
    3-D saddle cycles.
--- Point out stable limit cycles are structurally
    stable! First look at Poincare map.
--- Examples using polar coordinates. p35

Gradient systems.   ----------------------------------
Liapunov functions. ----------------------------------
Dulac's criterion.  ----------------------------------

Clocks: % ------------------------ This was done in earlier lectures
Explain mechanics.
From grand-father's clocks, via British Navy, to pocket watches,
to wrist watches, to bulova, to crystal, to atomic + internet.
Smaller the vibrator, the more accurate. Why:
--- easier to keep stuff unperturbed.
--- simpler [pendulum is messy, crystal is simple].
--- faster oscillations, smaller time units, hence potentially more accurate.
--- frequency more "rigid" [when phase locking into electronics,
    it is the electronics that does the job].

Idea behind electronic watches:
1 - Start with an electronic self-oscillator. Can be made very small, but they
    are not very accurate [hard to make e-components very precise].
2 - Couple it to an oscillator with a very "rigid and accurate" frequency.
    Example: small tuning fork [Bulova watches] or, even better, a crystal.
    [Coupling done magnetic or pyzo-electric].
3-  Oscillators lock, with frequency determined by tuning fork/crystal. Result
    is a very accurate self-oscillator.

Issue: how to output time? When the oscillator is very small, there is no
power to move anything, as in old-style mechanical watches.
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