Lecture 20 - Tue 2020 11 10 - Virtual
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Left to the students:
READ THE EXAMPLE of a driven-damped pendulum in section 8.5 book.
The example involves three bifurcations of limit cycles [Hopf, Homo-clinic, and
infinite period], and illustrates the use of the Poincare Map.
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% Flows in the 2-Torus. % ---------------------------------------------------- %
  Torus as square with glued opposite sides.

Simple example \dot{theta_j} = omega_j;  j=1, 2.
Show periodicity if omega_1/omega_2 is rational.
   Trefoils when omega_1/omega_2 = 2/3 or 3/2
   [project Torus on a plane, along main axis].
Case omega_1/omega_2 irrational. Orbit is ergodic.
   Precessing trefoil when omega_1/omega_2 close to 2/3

Example with nonlinear coupling: 
      \dot{theta_1} = omega_1 + k_1*sin(theta_2-theta_1)
      \dot{theta_2} = omega_2 + k_2*sin(theta_1-theta_2)
Show phase locking and compute compromise frequency.

More generic couplings do not necessarily yield phase locking, but they can
still produce synchronization [same period, but phase difference wobbles].
  In theta_1 -- theta_2 phase plane orbits attracted to one where theta_2
  advances a multiple of 2*pi when theta_1 advances some other multiple
  [reminiscent of the picture for the periodic orbits in the driven-damped
  pendulum]. This way can produce "simple" periodic orbits, or trefoils,
  or ...
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ALMOST PERIODIC % ------------------------------------------------------------ %
Define almost periodic via "almost" repeatable.
There are several, not quite equivalent definitions.
Example:
  y = y(t) is almost periodic iff, for every epsilon, there is an
  "almost period" T > 0 such that |y(t+T) - y(t)| < epsilon.

  Relationship with Fourier series,
                         can write: y(t) = sum a_n exp(i*omega_n*t) [A]

  Quasi-periodic as case where Fourier frequencies are generated by a finite
  set of frequencies:
  Any of the omega's in [A] has the form   omega = \sum n_j Omega_j
  for some integers n_j and a finite set
  of frequencies Omega_1, ..., Omega_N.

  Note how y gets more and more complex as the number of base frequencies
  grow.
  Landau hypothesis on Turbulence as a series of Hopf-on-a-Hopf. Leading to
  turbulence as an almost periodic process. Note:
  problem 1: almost periodic leads to orbit separation that grows linearly,
             not exponentially.
  problem 2: such sequence of bifurcations, generally, cannot happen in a
             dissipative system. Will explain this later. *******************
  problem 3: spectrum of an almost periodic function is still discrete, not
             continuous. Will explain this later. ***************************

Landau's hypothesis, however, is related to the Ruelle Takens "route to chaos".
In this route the first Hopf-on-a-Hopf happens, creating a quasi-periodic sln.
with two periods [which tracks a 2-D Torus in phase space]. But when the third
frequency appears, instead of three period quasi-periodic --- with a 3-D Torus
associated; chaos ensues. Geometrically, the T2 torus "crumbles".

If time permits, point out relationship with "small divisors" and KAM theory:
Perturbing   \ddot{x} + omega^2*x = 0, i.e.:   \ddot{x} + omega^2*x = f(x),
 leads to a nonlinear oscillator
 with a single frequency [equal to omega in the infinitesimal amplitude limit].
 Think of Poincare-Lindstead.
However \ddot{x_1} + omega_1^2*x_1 = f(x_1, x_2)
        \ddot{x_2} + omega_2^2*x_2 = g(x_1, x_2)
generates all the frequencies n1*omega_1 + n2*omega_2
and if omega_1/omega_2 is irrational, arbitrarily close to resonance terms are
generated [which generate the "small divisors"].       The result is that some
solutions survive but most do not [KAM theory ... explain].
Note that the scenario above is "sort of" what happens beyond the two period
solution.
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