[sdiy] slightly ot: Chua circuit and chaos

ASSI Stromeko at compuserve.de
Tue Feb 11 20:19:09 CET 2003

On Monday 10 February 2003 12:31, Czech Martin wrote:
> I'm just reading a book a second time, just to make sure I
> understand. The statement of this book is shocking:
> there is no real chaos in physics, it is only the models
> that can act chaotic. This topic was covered before,
> even by the "inventor" of numerical chaos, Lorentz.

That statement is profoundly wrong if it indeed appears in such 
generality. It is of course the case that modeling assumptions and 
nonlinearities introduced by discretization and finite range that are 
inherent to numerical simulations can lead to models that exhibit 
behaviour that does notoccur in reality. Note that this does apply 
generally, not just to chaotic behaviour. It is also true that with 
each new theory there's a bunch of folks running around telling that it 
either invalidates everything that was know before or that it explains 
everything or both. You don't have to wait for long for the rebuttals 
and some of them invariably throw out the baby with the bathwater.

The complete term for the type of chaos we're presumably talking about 
is incidentally deterministic chaos, which means (I'm simplifying) that 
the models are completely predictable, but any long-term output is not. 
So in a sense it is exactly the other way around: the models are not 
chaotic, but the system still is.

> I think the autor (Mr. Wehr) has done a good job
> showing how parasites take up a new fashion of science
> and draw more then speculative statements out of it.

That always happens. Look a few years back for instance to read the 
most outrageous claims about how everything but fuzzy logic and neural 
networks is obsolete.

> He also shows what huge methodical questions come up
> if physics takes place in the computer and not
> in the lab any more. Questions like:
> can we trust computer simulations, if different operation systems
> and hardware bring very different results for the same problem?
> Are real numbers a correct model for physicl quantities?

That are all valid questions that any competent modeler will have to 
answer, but this does not apply to the claim that chaos does not exist 
in reality.

Specifically, take a look at what is known as Chua's circuit. I've had 
the fortune to hear a guest lecture by Leon Chua on the topic and he 
did not make any undue claims about it, btw. The significance of Chua's 
circuit is that for a while it was believed that only inherently 
nonlinear systems would exhibit chaos - this circuit proved that 
conjecture wrong: it is piecewise linear. That made it possible to 
prove mathematically that the chaos it produces is not some artifact of 
residual nonlinearities, but it is also exhibited by the ideal circuit. 
In fact, the determinism is strong enough that one is able to produce 
two different circuits producing the same output (in a sense). Looking 
at the signal with various statistical tools you'd conclude that the 
output is noise, but if you take the twin circuit you can completely 
correlate the signal (this idea was once publicized as a means to 
"encrypt" electronic signals).

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