Lecture 16 - Tue 2020 10 27 - Virtual
% ==============================================================================
The Poincar\'e-Linstead method to look for periodic solutions.

More accurate than two timing if you want periods and frequencies accurately,
but it only gives you the periodic solutions [not their stability, nor how
solutions near them behave, as two timing does].

Objective: Looking for periodic orbits near something like a center.
Basically, we have an equation with periodic solutions that we can compute,
and want to compute periodic solutions for a perturbed equation.

Motivation: If the solution is periodic, we should be able to expand it in
Fourier Series, ... but we do not know the period [as the nonlinearity
affects it]. Thus must expand the period as well.

Method originates with Astronomers computing orbits of planets and such with
great accuracy. Explain: Solar system is, basically, a two body system [which
we can solve exactly] with a whole bunch of small perturbations caused by the
non-Jovian planets.

Example: Duffing's equation.  [See Web page notes on Weakly Nonlinear
                               Oscillators]. Cover pp. 3--6 of these notes.
STUDENTS SHOULD READ the other examples in these notes  (e.g.: Limit cycle for
van der Pol)!

% ---------------------------------------------------------------------------- %
  IMPORTANT QUESTION BY A STUDENT, and answer:
  Can this be done if the "leading order" problem is nonlinear?
  Answer: Yes, as long as you know enough about the leading order problem.

  Example (sketch only). Consider \ddot{x} + V'(x) = epsilon*f(x, \dot{x})
          where V is some potential with a local minimum, and we restric
          our attention to the neighborhood of the local minimum.

  Then \ddot{x_0} + V'(x_0) = 0, and you can write the solution in the
  way
         x_0 = U(T, E); T = omega t, and U 2*pi periodic in T.
  where E is the energy of the solution and omega = omega_0(E).
                                         (show how to do this).
  Then, for Poincare-Lindstead  omega = omega_0 + epsilon omega_1 + ...
  where the problems to solve have the form
    \ddot{x_n} + V''(x_0) x_n = G_n  [G_n = lower order terms].
  Because x_n has to be 2*pi periodic in T, G_n must satisfy the condition
    \int_0^2*pi u*G_n dT = 0.                         [B]
  for all u's that solve \ddot{u} + V''(x_0) u = 0.   [A]
  Fredholm alternative <----- Explained in the lecture.
  
  There are at most two linearly independent solutions to [A]. So, two cases:
    Single linear independent solution: in the perturbed system get a center.
           Only one condition at each n, which determines omega_n.
           But amplitude is arbitrary.
    Two linearly independent solutions: two conditions per n, get a limit
           cycle. At each step corrections to the frequency and the amplitude
           follow. Example: van der Pol.
           Exception: one of the conditions is satisfied automatically, by
                      some symmetry. Example: Duffin's equation.
%
% ---------------------------------------------------------------------------- %

Method of averaging: we will skip it in the lectures. You are expected to read
it from the book [it is not included in the Weakly Nonlinear Oscillator notes].

NOTE: the series in the small parameter "epsilon" that result from the various
   expansion methods that we have seen are (often) asymptotic, not convergent.
To learn the difference between convergent and asymptotic [and what asymptotic
means] *READ* the notes "Asymptotic versus convergent", posted in the web page.
%
% ==============================================================================
EOF