[sdiy] slightly ot: Chua circuit and chaos

[Richard Wentk]

At 14:38 11/02/2003 +0100, Czech Martin wrote:
> >>We simply use these numbers for every day engineering.
> >>Are we allowed to do so?
> >>Is the concept of real numbers a good model for what's
> >>really going on?
>
> >This will be covered by any decent university comp sci course. It's usually
> >called something like 'computing with approximate numbers.'
> >
> >You can't learn scientific programming to degree level without being aware
> >of the limitations of only having a certain level of accuracy. That's
> >certainly true for DSP programming as well as other things.
>
>The above statement refers not to limited accuracy in computers.
>The question is, if real numbers are used in all their infinity,
>are they a good model for certain systems?
>Charge is represented by a natural (whole) number of electrons or ions
>and so on.

But the two questions are synonymous if you use computers predictively.

The only real question is whether or not a method gets an answer that 
accurately reflects what happens in the real world. Charge may well be 
discrete, but it's discreteness is invisible and irrelevant in all but the 
most extremely sensitive physical electronic systems - in the same way that 
there are no practical situations where it makes more sense to treat an 
ocean as a collection of individual water molecules than a fluid with 
certain properties like viscosity, mass per unit volume, and so on. Or a 
pendulum as a collection of iron molecules, rather than an example of 
damped harmonic motion.

>As you say, "chaos" seems to collapse if real numbers are not
>used. Insofar it is allowed to ask if chaos really exists
>outside mathematical models (where it exists for sure).

No, chaos only *appears* to collapse because you're not including enough 
precision to do predictive calculations properly. Remember, chaos is 
recursive, and if there isn't enough precision to allow for small 
variations then chaotic effects can disappear.

The original point about population dynamics was that DEs model them 
accurately. where a numerical approach based on small integers won't. 
That's because what matters aren't the individual numbers, but their 
statistical distribution.

It's this fact that the author seems to be missing.

>systems which behave with natural numbers (e.g. number of electrons),
>are translated into D.E. with real numbers,
>these are solved using rational numbers as computer approximations
>of real numbers and difference equations as approximations
>of D.E., and finally this is believed to approximate the
>real world problem. So no wonder that this sometimes does not work.

I'd need to see some applications where this turns out to be true to be 
convinced by this.

>As far as "chaotic" or nonlinear dynamics are involved, I have
>never seen a basic course on numeric computation discussing these
>issues.

Try the e-print site at www.arxiv.org, which is where front line science is 
done. You'll find hundreds of practical applications of chaos theory there, 
covering every discipline from molecular biology to the dynamics of galaxy 
formation.

Richard