there is a lot that is getting going in the lab..

-earlier issues about not being able to recreate the attractor
from the time delay embedding;RESOLVED:
the issue was that the system was being under sampled.
so for instance, the data that i showed in a previous post (post 4) was under sampled.

here is one second of the data from the system measured on august 16 with a resolution of 1khz (1000 measurements per second)

figure 1

here is one second of the same system measured later, at 10khz (10,000 measurements per second).

figure 2

there is clearly a large difference. let us look closer at the later(high resolution) data-

figure 3

this is the chaotic behavior of the system. with this data, i am able to reconstruct the attractor of the system (via time delay embedding) using only one measurement (x) , instead of two (x and y). FINALLY! not only that, but, since i can create as many time delay variables as i want, i can embed the data in arbitrarily high dimensions. for instance, if someone wanted to plot the attractor in three dimesions without time delay embedding (or some other equivalent trick), they would have to setup a whole new measurement on the system.
with time delay embedding i just have to construct a new time delay variable. for instance, from the two-dimensional embedding x(n), x(n+t), i could go further into three-dimensions by introducing x(n+2t).
so, here is a successful three-dimensional reconstruction from time delay embedding.

figure 4

i wish i could post it such that you could rotate the thing in 3-D.. it would be nice.. but that (though possible) would likely take a while. i can rotate it in 3-D.. and it is really cool..


[what is funny is that, if you plot under-sampled points from output x and y (as seen in post 4, figure 6) you can still recreate the attractor. this is because the two variables x and y stay correlated over time, and so, no matter when you sample them, the pair of their values will lie on the attractor.
however, if you take the variable x, and plot it against a time delayed copy of itself (see post 4), then the sampling has an enormous impact on the relationship between the variables (since they are time dependent constructs). so, the absence of structure in post 4, fig. 8 results from the system being under sampled, and the measurements being so far apart is equivalent to improperly choosing the time delay really really big.. this causes the variable x(n) and x(n+t) to become uncorrelated, so that the time delay reconstruction gives no information..

the fact that the attractor was still being recovered even though the dynamics were not being sampled well was initially how i convinced myself that there was not a sampling problem. ]

the photo below is alex tozzo, a colleague here. he is awesome. he sort of makes sure that everything that we need is in place and working. though he is a mechanical engineer, he is one of the resident chaotic circuit caretakers...
here he is, toiling in the gumption trap that is a nonlinear thing which is not doing what you want it to do..
(the lens cover of my camera does not open by itself all the way.. and it has to be opened with the fingers. i sometimes forget.. )

figure 5

a week or so ago, we received two high school interns in the lab.
gabe and chirag.
chirag has become interested in the chaos work that we are doing, and so he will be jumping into the project with us. he is quite virtuosic with computers.. below is a photo of a few labpersons (left to right: chin, frank, chirag, alex). chin bet frank lunch that chirag could not get the data acquisition scheme working...

figure 6

here he is below, having just worked out the data acquisition scheme .. now we can take data from multiple sources, at more than sufficient resolutions, for very long durations. CHOICE!

figure 7

this following audio is from the chaotic circuit itself. alex and company rigged the audio output last year sometime while using synchronized chaotic circuits to mask (kind of encrypt) an audio signal..
enjoy..
i think that alex's phone rang (silently) right at second 15 or so.. .which is when the frequency moves a little higher.. and maintains consistently higher.

CHAOSaudio by a philadelphia experiment


this is just the first audio interpretation of chaos (hopefully). chirag has written some code that takes text musical notation and converts them to midi outputs.. so now, all i have to do is alter some of my code so that i can take chaotic sequences, and convert them to text denoting musical notes with duration, and then together, we will make chaotic music..
more to come