Lecture 15 - Thu 2020 10 22 - Virtual
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A) Reduction to one dimensional systems.
   The same idea we used to construct the limit cycle in the mu >> 1 limit
can be used to justify the reduction to a 1-D system for highly damped
situations. READ THE NOTES "Reduction to 1-D systems" in the web page.
            ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
B) In the next lectures we will cover perturbation methods:
    Poincare-Lindstead, Two-Timing/Multiple Time Scales, and Averaging,
    as well as the meaning of "asymptotic".
For extra details see the "Weakly nonlinear oscillators" notes in the web:
http://math.mit.edu/classes/18.385/Notes/weaklyNLoscilators.pdf
Strogatz book also has something, but it is a bit limited.

We will illustrate the methods using the van der Pol equation, written in
the form
                        \ddot{x} - \epsilon*\nu*(1-x^2)*\dot{x} + x = 0,
where epsilon is small
and \nu = \pm 1. This allows us to look at the cases where
\mu = \epsilon*\nu is small and both positive or negative.
We will also look at
Duffing's equation      \ddot{x} + x + \epsilon \nu x^3 = 0.

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Method of two times (multiple scales) for van der Pol.
1) "Geometrical" motivation. Why does this work? What is the idea?

   When \epsilon is small the solution should stay close to the unperturbed
   problem for any "finite" time interval, but "which" solution it is close
   to can drift over long time intervals, because of the small nonlinearity.
   Hence look for a solution where the parameters of the unperturbed
   problem change slowly,  adapting to track the "local" unperturbed
   solution.
   Example: if the unperturbed solution is a*cos(omega*t), then we look for
            an approximation that has the form  a(\tau)*cos(omega*t), where
   \tau is a suitable slow variable.  For example:   \tau = \epsilon*t,  or
   \tau = \epsilon^2*t, or ...  --- determined by how fast the nonlinearity
   changes the solution.

   In addition, while the solution "stays close to an unperturbed solution",
   close is not the same as equal.   For example, the unperturbed solutions
   may track circles in the phase plane,  but the full solution will not do
   so exactly.  So, we also need to add "higher order" shape corrections to
   the leading order approximation a(\tau)*cos(omega*t).

   Let us do now an example:  go through pp. 9--11 of the "Weakly nonlinear
   oscillators" notes.

2) NOTE: the same argument motivates the "averaging" method. If the solution
   satisfies "x \approx a(\tau)*cos(omega*t)",  we can average over the fast
   phase theta = omega*t to obtain equations for the slow variation.
   
   I will not cover the averaging method in the lectures.
   READ IT IN THE BOOK. *********************************

3) READ the notes: "The slow time is NEEDED" in the web page.
   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
   These notes show an example of what happens if you do not introduce a slow
   time.  Then the actual nonlinear solution drifts away from the unperturbed
   one, which causes an error that grows steadily [secular term],  and limits
   the validity of the approximation to short times. WORSE, the approximation
   completely fails to capture the effect of the nonlinear term;   just about
   the only information it carries is that
         "for short times the solution looks like the unperturbed one".
   This is useless! --- you do not need to do a calculation to know this. The
   whole point of a perturbation expansion is to capture the effect caused by
   the small extra term. Else, why do it?
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