Lecture 15 - Tue 2024 10 29
Relaxation oscillations, two-time scales.
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                                         RELAXATION OSCILLATIONS for van der Pol.
Where I left last lecture:   \ddot{x} - mu (1-x^2)\dot{x} + x = 0.
Let x = \dot{z}, then:       \ddot{z} - mu (1- (1/3)\dot{z}^2)\dot{z} + z = 0.
Let
   u = \dot{z}, v = z/mu. Then:  \dot{u} = mu (u - u^3/3 - v), \dot{v} = u/mu.

Alternative: write van der Pol 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 too high. Reason "expensive" amplifiers use many stages.

Calculate approximate period in this limit [see last page of the notes
"van_der_Pol_existence_and_period.pdf"]
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                               Relaxation oscillations and stick-slip phenomena.
Give a physical examples before the math: stick-shift motion
          Nasty noise by chalk on board, noise when dragging a chair,
          refrigerator pushing, etc.
If time permits do the example in the notes,
            "Relaxation_oscillations_due_to_friction_stick_slip.pdf"
      or at least point out that the reason is the shape of the friction
      versus velocity diagram, and how it works.
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EXAMPLE:                                                Done in a prior lecture
Justify dropping inertia for   eps \ddot{x} + \dot{x} + x = 0.
Follows the same idea, write equation as:  \dot{x} = v and \dot{v} = -(v+x)/eps
and use fast scale in v versus slower scale in x.
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                                                                Two-time scales

Explain idea, using vdp for small gain:   \ddot{x} + x + eps (x^2-1)\dot{x} = 0.
"Locally" in time [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. 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/eps [see #1 below]. Hence
we allow a and b to be slow functions:          a = a(tau), b = b(tau)
where tau = eps t. Thus the
approximation becomes          x ~ a(tau) cos(t) + b(tau) sin(t) +
                                   eps x_1(t, tau) + eps^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 the order in the
approximation a resonance first occurs.  This will be explained later, after
we see an example. It is also explained in the web page notes cited above.
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