Lecture 16 - Thu 2023 11 02
Relaxation oscillations.
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                                                        RELAXATION OSCILLATIONS.
We will look at the limit cycle for van der Pol both in the limit of a small
and large gain. First the large one
               \ddot{x} + mu (x^2-1)\dot{x} + x = 0; mu >> 1.
Write it as  \dot(\dot{x} + mu*(x^3/3 -x)) + x = 0

Define  mu*y = - \dot{x} - mu*(x^3/3 -x);
so that:             \dot{x} = \mu*(x - x^3/3 - y)
                     \dot{y} = x/mu

Show how this leads to the relaxation oscillation limit cycle.
Sketch how x = x(t) looks like. Note the time scales 1/mu and mu.
       Period is O(mu)

Note how nonlinear the oscillations are, with lots of high harmonics,
     as opposed to the near sinusoidal behavior for mu small.
This signal would sound very bad if passed through speakers.
This is related to the screechy noises in electronic amplifiers when gain
     is cranked out high. Reason "expensive" amplifiers use many stages.
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Relaxation oscillations in mechanical systems: stick-slip.

Give a physical example before the math: stick-shift motion ==> Nasty noise by
  chalk on board, noise when dragging a chair, refrigerator pushing, etc.
Maybe do the example in the notes,
      or at least point out that the reason is the shape of the friction
      versus velocity diagram [explain why it is so].
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EXAMPLE:                                                Done in a prior lecture
Justify dropping inertia for   eps \ddot{x} + \dot{x} + x = 0.
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                                                                Two-time scales
Explain the idea, using vdp for small gain:
        \ddot{x} + x + eps (x^2-1)\dot{x} = 0.

"Locally" in time [some finite O(1) period] any solution can be approximated
by a solution to the "unperturbed" equation
                                            x \approx a cos(t) + b sin(t);
for some constants a and b. However, for
different time periods, we cannot use the same constants. Hence we need to
allow these constants to slowly evolve, to adapt and be "always" the "right"
ones.
The time scale for the evolution of these constants is roughly given by the size
of the perturbation. That is, the solution cannot change by O(1) over periods
shorter than 1/epsilon; [see #1]
hence we allow a and b to be slow functions:   a = a(tau), b = b(tau)
where tau = epsilon t.
Thus the approximation becomes  x ~ a(tau) cos(t) + b(tau) sin(t) +
                                    epsilon x_1(t, tau) + epsilon^2 x_2(t, tau) +
                                    ...
where the higher order terms allow for corrections to the shape of the trajectory
in phase space [i.e.: even for a finite time, the trajectory is not going to be
a perfect arc of a circle]. 
                                WE WILL DO AN EXAMPLE NEXT LECTURE

More details in the following notes in the web page:  READ THEM!
Weakly Nonlinear Oscilators [specially section 2].
Hopf Bifurcations [specially 1.4, 1.5, 1.6, and 1.7].
Hopf bifurcations using two timing and complex notation.

#1 But the time scale for changes could be slower [depends on the details of
   the perturbation]. Mathematically this is determined by at which order in
the approximation a resonance first occurs. See web page notes.
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