[FRIAM] Fractal discussion Landscape-bird songs

[Jon Zingale]

Nick,

Well one way we may be able to understand
birdsong as fractal might be by studying the
underlying mechanism of the syrinx
<https://en.wikipedia.org/wiki/Syrinx_(bird_anatomy)>. I can
imagine this section of the birds trachea as a coupled
oscillator, that when driven far from equilibrium
could give way to trajectories along a strange
attractor (which would be fractal). In an attempt
to think about recovering the attractor from the
time-series of the bird song, I ran across Takens'
theorem last night. Then later last night (I couldn't
sleep) I coded up an example of Takens' theorem
in RubyProcessing
<https://github.com/jonzingale/RubyProcessing/blob/600c83727c77a4d52ac4effe65d7258100bae5b4/lorenz/lorenz_reconstruction.rb>.
What is amazing about this
theorem is that it suggests how to build a low-
dimensional manifold from a single dimensional
time-series! So freaking cool. As a test case, I
coded up the Lorenz equations and plotted the
manifold. Then I calculated just the time series
for the x dimension. Lastly, I reconstructed the
entire manifold (topologically) from just this one
coordinate! Included below is a screenshot of
the visualizer. It is actually more fun to watch in
motion, but the picture is telling in itself.

Jon
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