From: "Tim Stinchcombe <timothy@t...>" <timothy@t...>
Date: Fri Jan 10, 2003 12:00 am
Subject: Re: chaos module

Hi Tim,

I know the chaos theory, I have a book on it. It's fascinating
subject, but only until recently I didn't know that they
translated the theory in a famous/notorious module!
I just never heard of those oscillator's you named.

I also went to a lecture on this subject and it was
very clearly explained by this professor. I know the chaos
theory orginates from theorectical physics. He had
a simple model with him and demonstrated the how chaos is
created. He had only a swing which had two wheels attached to it.
First when he moved the swing, it made preditable moves,
but beyond a certain point, totally different things happened
and amazing too see, you couldn't predict any of this
strange movements anymore. This was the edge of chaos.

Chaos travels from the center to the edge, where it reaches
a point that totally new phenonomen occur. Like your famous
sample of the butterfly flapping it's wings in North America
can cause a sandstorm in the Sahara.

Thanks for that great link. What a fine site with glass clear
pictures!!! I have benchmarked in my favourites and when I
have time and are up to it(oops those maths, but I have to be
cool and don't turn my back on them!).

Roel


Hi Roel,

> What the hell are a Chua oscillator and a Duffing oscillator
> Can you eat it
> I never heard of this before.
> Most intriquing, please explain!

Tim Wrote:

I have absolutely only a very rough grasp on any of this, but here
goes. Some of what chaos theory about is how a small change in one
part of a system can have a large affect on another (a favourite is
something about a butterfly flapping its wings having an affect half
way round the world...). It's possible to put together electronic
circuits that exhibit this kind of behaviour, i.e. a small change
will suddenly make the circuit jump to a completely different state.
Chua's circuit is a well known example (named after the guy who first
made it), Duffing is another, and they are both deceptively very
simple to look at. They oscillate in strange ways, and changing
component values by small amounts can completely change this. Chua's
circuit has many different 'modes' - if I can get the upload to work,
I'll stick a file in the files area with some plots. In it you'll see
what is called a 'double scroll', for obvious reasons. Altering the
pot value can make one half of the scroll disappear; at another it
goes much larger, called a 'limit cycle' (it sort of saturates). They
can be made to oscillate at audio frequencies, and the top one of the
pair looks rather like a square wave through a highly resonant
filter, and apparently sounds quite interesting. I have a paper where
they control the oscillator digitally in order to make some decent
sound (in tune I believe) from it.

The theory is very mathematical, with lots of heavy differential
equations in many variables. The 'thing' which make Chua's circuit
work is what the two op amps do - they emulate a negative resistor (V
over I through a normal resistor is of course R; V over I for this
thing gives -R, i.e. has negative gradient!).

And that pretty much sums up all I know. Basically it looks like
something fun to play with! Just remembered a good website to look at
is Dan Slater's page:

http://www.nearfield.com/~dan/Music/chaos/Chaosrel.htm

It'll explain it far better than I can, and includes stuff on Buchla
and references to several good papers.

Cheers,

Tim