[FRIAM] Fractal discussion Landscape-bird songs

[Robert Wall]

This less than 5-minute video seems visually helpful in the context of what
Jon accomplished last night: Takens' theorem in action for the Lorenz
chaotic attractor <https://www.youtube.com/watch?v=6i57udsPKms>.  This
video provides some idea of what emerges from the manifold in terms of the
time series with airflow. Yes, very cool! 😎

On Tue, Feb 28, 2017 at 4:22 PM, Jon Zingale <jonzingale at gmail.com> wrote:

> 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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