Lecture 13 - Thu 2020 10 15 - Virtual
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Continue with Limit cycles
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Recap definition (isolated periodic orbits) , and the fact that stable limit
cycles are structurally stable [contracting Poincare map].
Ruling out cycles
  Gradient systems    [Last lecture].
  Liapunov functions  [Last lecture].
  Dulac's criterion.  [Introduce and give example].
Showing a limit cycle exists
  Trapping regions [we will see this later].

% ---------------------------------- Clocks ---------------------------------- %
  This was done in earlier lectures, except for ## points.
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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The first and "basic" electronic self-oscillator [i.e.: limit-cycle] and the
math. model: The van der Pol equation
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Radio transmission and electronic oscillators. A story that came before
electronic watches [e-watches are just mini-versions of this,
miniaturization, not possible till late 50's ... but radio much earlier].

Van der Pol oscillator and vacuum tubes [same idea with transistors].
% --- This portion was done in earlier lectures
  Basic circuit: RLC ... damped oscillator, same as pendulum or swing.
  Need to feed energy to it: what is the e-analog of the "Anchor escapement"
       set up of mechanical watches [~1670] for e-oscillators?
  Describe diode and triods [vacuum tubes].
  Describe how to set it up so it kicks RCL circuit at right time.
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Resulting equation is [van der Pol, mid 1920's].
  To see a "complete" derivation of the equation, see
  REFERENCE:
  Stoker J.J. Nonlinear vibrations in mechanical and electrical
              systems. Wiley 1992.

The basic idea is that the feedback mechanism adds a term g(x)\dot{x} to
the damped harmonic oscillator. Then an expansion of g for "small" values
of x [the current] yields the equation:

Van der Pol oscillator	\ddot{x} - \mu*(1-x^2)*\dot{x} + x = 0.
For this equation
we will show:
--- Has stable limit cycle for all mu > 0.
--- Will inspect limit cycle for mu >> 1 and 0 < mu << 1.
--- mu >> 1 ... relaxation oscillations (bad for e-devises).
		analog of stick-shift oscillations
--- 0 < mu << ... nice, harmonic-like behavior.

To prove that a limit cycle exists for all mu > 0, we will use the idea of
trapping regions and the Poincare-Bendixon theorem
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% ----------------- Poincare Bendixon Theorem [review, stated in prior lectures]
Re-state it.
Point out it says: orbit limit is infinity, critical point, cycle graph,
    or PERIODIC ORBIT *not* limit cycle as the book  often implies. Note: with
    "probability one" a periodic orbit is a limit cycle [explain why], but an
    orbit approaching non-isolated periodic orbits is possible!
Examples to see all possibilities are easy to do with "radial systems"
    \dot{r} = f(r)   and   \dot{theta} = g(r, theta).

Note that PROVING that a periodic orbit is a limit cycle can be tough. You need
to show that it is isolated. Poincare-Bendixon does not do this, something like
Dulac's criteria needs to be used as well.
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