it has been an easy hectic bit, .. but i have returned..

getting down to it-
mathy stuff is in brackets [],

we have taken data from the chaotic circuit, which is quite gracious, compared to any circuits that i have ever met, in the way of behavior..
the "chua circuit" is basically a relatively simple circuit which is capable of many different types of behavior
figure 1

[it is a nonlinear dynamical system, deriving its nonlinearity from a cleverly constructed nonlinear element called the "chua diode".
if you would like to dive into it, this is for you].
put simply, the circuit has certain properties [parameters], which can be changed, in our case, we are changing the voltage being applied to the "nonlinear resistor", which changes its resistance in the chua diode.
for some particular values of the parameter, say, when applying 2.5 volts, a the circuit can have a perfectly repetitive [periodic] oscillating behavior. [you are looking at the measurement "x" is changing over time]
figure 2

then, if we were to decrease the voltage slightly, the behavior of the circuit becomes a little more complicated but still periodic- notice that the oscillations alternate tall-short-tall-short, yet, the behavior is still repetitive. (this is still x, beginning to behave more erratically)
figure 3

[the system exhibits a period doubling cascade]as we change the voltage further, the behavior becomes more "complex"
figure 4

until, at some particular voltage, the behavior becomes chaotic, that is, completely aperiodic (non-repeating).
figure 5

the variable y does the same thing.
if we plotted x against y, (looking at part of the system's state space) the previous progression looks like this;
figure 6

from the oscilloscope, which displays the measurements in real time, the system in chaotic behavior looks like this
figure 7

this shape is called the double scroll attractor. its spacial structure in state space corresponds to the dynamical structure of the chaotic behavior. it is called an attractor of the system. and since it is fractal, it is called a "strange attractor".


this is all great..
but the trouble comes when i try to implement my analysis, which consists of something called time delay embedding.
this is when you reconstruct the attractor of the system, with only one variable (say for instance, x), by "plotting it against itself (with a small time delay), instead of y. this is essentially defining a map from the data in x, to a higher dimensional space.
by the theory (and all of my experience with it), one may reconstruct the attractor in this manner.. yet, i am running into trouble using just the data from x. let x(n) be the value of x at time n. then, we "embed x with a time delay t", meaning we plot x(n) against x(n+t) . and we see that, the double scroll does not manifest,
figure 8

nor does it seem to manifest in higher dimensions (which is usually the trick)...
at least, i think this is a problem that i am having.
i still have a few things to check.
one thing is for sure,.. the measure of complexity that i am implementing is treating this embedded chaos as though it is random noise.. which suggests to me that either there is some information missing in the embedding (which could come from not embedding in a high enough dimension, or possibly relying on another variable and/or measurement conditions), because the measure should (and generally does) have a higher value on chaotic data than on random data. i know that our data is chaos.. so, i just have to convince my computer that it is.. and then things will get going.