Lecture 13 - Tue 2024 10 22
Continue with Hamiltonian Systems; Liapunov Functions; Limit cycles' etc.
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                                                           On clocks and watches.
A brief story of time keeping and how it relates to limit cycle
oscillators [the 2-D version of what driven-damped systems do], synchronization,
and clever designs for how to couple and drive oscillators to make clocks.
See the file "Links_to_Clock_stuff.rtf"
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                                      Correction to calculation in prior lecture.
In lecture 12 (towards the end), when calculating the approximate radius of the
limit cycle for the van der Pol oscillator
                                           \ddot{x} + \mu (x^2-1)\dot{x} + x = 0,
I ended up with the integral
                                \int_0^{2\pi} (a^2 cos(t) - 0.25 a^4 sin(2 t) dt,
and said that the integral of
the square of sinusoidal is 1/2.  This is wrong:  it is the average of the square
of a sinusoidal that is 1/2.   This means that in the calculation I did there was
a factor of 2 pi missing. It does not change the answer  (since for this the only
thing that mattered was when the integral vanishes)
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                                                     van der Pol important point.
As $mu$ grows, the sinusoidal character of the limit cycle is lost (do some
numerics using PHPLdemoB). Leter on we will see that for $mu$ is large the
oscillations become very "jerky", with sudden changes.

This has a bearing in electronics. Non-sinusoial sound waves, with lots of high
frequencies, sound very bad to the human ear.   Hence components in "HiFi" have
to work in the near sinusoidal regime, which is the reason why "many stages"
are needed in a HiFi amplifier. In the pre-microchip era, this made them very
expensive.
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                                                              Volume calculation.
For an arbitrary system \dot{Y} = F(Y) in n-dimensions, the equation
for the evolution of the volume of a differential blob in phase space
is:
     \dot{V} = \dive(F) ......  Showing this left to a problem set.
Examples:
Hamiltonian:
    \dive(F) = \sum (\dot{q_n})_{q_n} + (\dot{p_n})_{p_n}
             = \sum (H_{p_n})_{q_n} - (H_{q_n})_{p_n}     = 0.

     VOLUME PRESERVED. This is known as Liouville's theorem.

Gradient:
    \dive(F) =   \sum (\dot{x_n})_{x_n}
             = - \sum (V{x_n})_x_n       = - Laplacian V

    Thus volume decreases near any local minimum, or everywhere if Laplacian V > 0.
         volume increases near any local maximum, or everywhere if Laplacian V < 0.
         Near a saddle it can expand or, depending on the relative sizes of the
              positive/negative eigenvalue.

Mixed Hamiltonian-Gradient systems       \dot{q_n} =   H_{p_n} - Phi_{q_n}
                                         \dot{p_n} = - H_{q_n} - Phi_{p_n}

Then  dive(F) = - Laplacian Phi [the gradient part controls the volume evolution]

In "general", dissipative physical systems contract volume. Hence, quite often,
in dynamical systems dissipation is associated with volume loss.  But, as with
all "mathematical" generalizations, you have to be careful with this.

Example: write van der Pol as a system
               \dot{x} = v   and   \dot{v} = mu*(1-x^2)*v - x

Then dive(V) = mu*(1-x^2).
     The volume increases for x^2 < 1 [gain        dominated region].
     The volume decreases for x^2 > 1 [dissipation dominated region].
"Surprising" fact:
 Any disk that does not include the origin has its area --> 0 as t --> \infty
 If the disk contains the origin, then             area --> A as t --> \infty
 where A > 0 is constant.
 QUESTION TO YOU: Why, and what exactly is A?
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