[sdiy] What is chaos?

[John Richetta]

My understanding is that what Paul and Ian have written is correct.

But note this important point: chaotic systems are, essentially 
always, systems with feedback, and an amplifying regime which also 
exhibits some nonlinear behavior.  This input-to-output 
amplification, feedback, and nonlinearity may be deterministic, but 
it is also extraordinarily sensitive to changes in initial 
conditions.  I think "extraordinarily" in many cases actually means 
"infinitely."  Infinity is a big multiplier. :)

So, while it is true that deterministic computer simulations (the 
vast majority ;) ) may be reproduced exactly, it is *not* necessarily 
true that a computer system is good at simulating weather.  The total 
ecological system that constitutes Earth weather appears to be 
chaotic, and so is very sensitive to changes, of extremely small 
magnitude.  The reasons that even fast computer simulations fail to 
track actual weather accurately for long periods include:

    1. imperfect models
    2. incomplete initial data
    3. inaccurate initial data
    4. limited resolution in simulation math

While it is possible to correct some of these to varying extents, 
note that because of the finite precision of computer math, we expect 
that, if the universe is truly analog (unclear at this point), then 
it will probably never be possible to make a perfect simulation. 
This is because the high sensitivity to small deviations from 
reality, in the model or its data, means that eventually, any 
simulation, no matter how good, will tend to drift, and eventually 
the error drowns out the rest of the solution.  (Then, what you have 
is a good simulation producing a useless result.)

This is not something that will be fixed by, say, doubling the 
precision (though it will improve the result, certainly - by roughly 
doubling the useful simulation time).  Any error, regardless of 
source, is amplified.  That's the essential point.  Since computer 
arithmetic is far from perfect, any computer system will introduce 
deviations from the correct result that will be magnified.  Thus, 
sooner or later, results produced by the system will be incorrect, 
even if the original data was perfect.  (Again, this assumes the 
physical weather system has infinite analog precision, requiring the 
same of the model.)

Another way to say this, that some might not like, is that digital 
math is actually noisy, just like the analog systems we so readily 
see as noisy.  The operations themselves are not noisy, and digital 
systems are nondeterministic, but their ability to approximate the 
mathematical ideal is, from a mathematical viewpoint, hopeless crude 
(that is, finite, rather than than infinitely precise).

Anyway, chaotic systems are underutilized in synthesis, including 
analog circuitry, IMO.  This is an area I've been exploring casually, 
and have some ideas about, though it would be naive to call them 
truly original.  If you want to experiment, build a feedback system 
with some gain and nonlinearity (and what circuit isn't nonlinear?): 
you can usually elicit chaos from such a circuit.  Of course, that 
doesn't mean all circuits are equally chaotic.  My usual setup for 
experimentation includes sample-and-hold circuits (to control the 
rate of evolution of the chaos - crudely, its "frequency").

One thing I'm trying digitally (oops, off topic!) is to achieve a 
"steerable" chaotic instrument: I like the idea of being able to 
selectively introduce chaos, but control the amount, to some extent. 
While mixing might qualify as one way, it's not very interesting to 
me.  I'm more interested in trying to map out various chaotic 
regimes, and let performers "steer" between them, veering into orbits 
that are fairly stable, or less so, as desired, by perturbing the 
system by small amounts, at critical points, in various directions). 
I think this is a very interesting controller, for a wide-range of 
musical parameters, including, dare I say it, direct listening 
(sometimes).  I would assume this could be done in analog circuitry, 
but based on the approaches I'm using right now, it's not obvious how 
it could be done well, or easily.  Others may have good ideas how to 
tackle this.

BTW, as long as we're discussing chaos, I can't resist relating 
another interesting phenomena that some may not be aware of.  I don't 
know the extent of the generality of this, but under some conditions, 
if you superimpose a signal on two chaotic streams, then, even if 
they drift somewhat from one another, you can usually substantially 
recover the signal, assuming the two chaotic stream are generated 
using pretty much the same chaotic system rules and starting state.

To make this a bit more concrete, I'll recount the example used to 
illustrate this to me: a voice signal was added to a chaotic signal. 
Listening to the result, you heard, well, chaos.  Another "closely 
matched" chaotic generator was subtracted from the combined signal, 
and the result was intelligible.  I think this works because of 
significant spectral coherence in speech, and the brain's facility at 
detecting those patterns, and ignoring what presumably is a large 
noise component in the result.

Although this result was surprising to some, it actually seems 
reasonably intuitive to me.  I think the system works because the 
noise component, which is the difference in the chaotic generators' 
output, is modest, in relation to the other signal, and often either 
somewhat periodic, or else fairly broad spectrum noise, both of which 
can be separated or ignored by the human auditory system (not to 
mention other forms of processing).

I believe some sort of synchronization was used between sender and 
receiver, to ensure that the drift of the two chaotic generators was 
minimized.  Note that in a digital system, drift can be easily 
reduced to zero.  It might be worthwhile experimenting with 
deliberately misaligned generators; conceivably, these produce types 
of noise and partially chaotic and periodic structure that are also 
useful (per above, I think it will be bursts of noise, similar to 
original, plus periods of evolving slightly chaotic periodicity). 
Hmm, an experiment I hadn't thought of until now!  If someone tries 
it, kindly share a taste of the results.  (Another layer of possible 
interest: modulate the degree of mismatch of the generators.)

Anyway, I'm done blathering.  I hope this was mildly interesting. :)

-jar

P.S. - If someone can point me to more info about that experiment of 
information transmission using chaos, I'd appreciate it.


At 9:33 AM +1100 2/13/07, Paul Perry wrote:
>It is true that chaos functions follow deterministic rules,
>and therefore cannot be called 'random'.
>And so, a chaotic function generated by a digital computer
>gives the same result each time......
>but, in the analog world, this is not so.
>
>Because, in the presence of noise - no matter how little -
>a chaotic system will diverge radically from the 'noiseless'
>predicted pattern. This is an essential property of a
>chaotic system. Which has obvious relevance to e-music.