Lecture 14 - Tue 2020 10 20 - Virtual
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Continue with Limit cycles. Show they exist using trapping regions.
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Van der Pol oscillator	\ddot{x} - \mu*(1-x^2)*\dot{x} + x = 0.
This equation
--- 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 [see review in prior
lecture].
Note: another example is given in Strogatz book (glycolisis, how
      oscillations happen in biology).
      PLEASE READ THE EXAMPLE OF GLYCOLISIS IN THE BOOK.

Note: in the Lecture I will only describe the general idea. For details
      see the notes "van der Pol trapping and relaxation osc. period"
      in the course web page. [right next to lecture 14]. READ THESE NOTES.

Step #1:
Inspect equation for the "energy" E = 1/2 (\dot{x}^2 + x^2).
Step #2:
Construct a trapping region for the van der Pol equation, showing that a
periodic orbit exists for all mu > 0. The proof that this periodic orbit
is actually an attracting limit cycle will not be done.    This requires
showing that the periodic orbit is unique  [can be done using a modified
version of Dulac's criteria].     Then everything in the trapping region
must approach the single periodic orbit there.

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% -------------------- LIENARD SYSTEMS [defined in the book] ----------------- %
  are a generalization of
  the van der Pol equation:    \ddot{x} - \dot{f(x)} + V^\prime(x) = 0,

where f = f(x) is a function that               "looks like" x - (1/3)*x^3
and   V = V(x) is a single well potential that  "looks like   (1/2)*x^2
[for precise definitions see Stoker's book].
These systems all have a unique attracting limit cycle -- same proof.
Note: in the derivation of van der Pol, f(x) = x - (1/3)*x^3 is an
      approximation, resulting from Taylor expanding.


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% ------- Relaxation oscillations. The case mu >> 1 for van der Pol. --------- %

Use "Lienard transformation" and re-write system [integrate equation via
   x = \dot{z}, and then scale]. Note book does this a bit different.
   Idea is to reduce system to form  \dot{x} = (1/mu) something and
   \dot{y} = mu*h(x, y), so equation reduces to h(x, y) ~ 0.

Analyze now system for mu >> 1. Find limit cycle.
Plot how solution x = x(t) looks as a function of t, with the fast transitions
  O(1/mu) in time, versus O(mu) for the slow parts.

Calculate period of the limit cycle for mu >> 1.

Explain physics: capacitor charges slowly, with sudden discharge at some
 critical value. Typical of "relaxation" oscillations.

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Mechanical example: Stick-shift phenomenon in friction.
 Illustrate it with pushing table/chair with/without big weight.
 Energy accumulates in elastic forces, with sudden discharge into kinetic
 energy at some critical value, and cycle starts again.

 See "Relaxation oscillations from friction" in the course web page.
 I will skip this in the lectures, but I expect you to READ THEM.

 This notes deal with the  example of a "mass-on-a-belt-and-a-spring". They
 derive the equations and show that they lead to relaxation oscillations in
 the limit of a weak spring [or a heavy mass].
 They get the appropriate non-dim parameter that characterizes this.
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