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 <jonzing...@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 > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove >
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove