Neato chaotic equations for analog computers to display?

Three very loosely coupled oscillators of similar frequencies. Will they lock to common frequency or will they be chaotic? --

Boris Mohar

wrote

The temperature, Down Jones average, height of blades of grass ...

All constrained, cyclic and unpredictable.

It's not chaos that's the odd duck in the universe, the odd duck is deterministic mathematics. There is _nothing_ in this universe that repeats itself, let me repeat that...

"God created real numbers, Man made the integers",

- Somebody famous, if you move in the right circles, but I forgot his name.

"And the Devil made 0 and infinity"

- Me, but I am sure Somebody Else wrote it down first

-- Post brought to you by Me, Somebody and Somebody Else

Here's another implementation:

Why don't you try to design Chua's chaotic circuit ? Firstly it is simple, further it does not require multipliers and thirdly someone has already performed 'Music Signals from Chua's circuit' , something which also seems to be your aim. I am sure you will have fun another weekend also :)

Well different sound from different spiral can be a nice option to study in Chua's circuit as well. I don't know if anyone has done that.

You can get reference on Chua's circuit from

I said simplistic... but in the meantime I will endeavor to hook up a rheostat under my desk that I can use to change the parameters driving the stock market :-).

Tim.

The temperature? Maybe since it's connected to the weather system which can display chaotic behaviors, but it's VERY high dimensional. That would be a BIG circuit.

Dow Jones? Nah. Best modeled as a stochastic system. If you think the weather is high dimensional...

Blades of Grass? Huh? I don't get that one. Fractal, maybe? Maybe, not. Connection to chaos or dynamics?

Being Cyclic is not a necessary trait of chaos.

Look (google) for simpler systems. Check out papers by Tom Carroll (in the 1990's), Sprott (1990's and later), Rulkov (late 1990's paper on just your topic), and the double-scroll circuit (often mislabeled as Chua's circuit, but that latter label has stuck).

-- Lou Pecora (my views are my own)

Maybe for audio you could use frequency, volume and left-right stereo pan as output parameters, maybe add a voltage-controlled filter fed with white noise instead of a pure tone.

What about feeding 3 outputs to an RGB LED

has anyone created a nonlinear/chaotic circuit in SPICE? i am interested in creating a model of a Hopfield Network in hardware having written a training algorithm for one. any takers?

ministry

That doesn't make sense. It's the temperature of 'something.' What's the something?

What you mean here is bounded. The motion is bounded. And you are correct that without that all trajectories might diverge to infinity.

The word 'cyclic' in most science and engineering usually means periodic or nearly so or repeating. Chaotic motion does have periodic motions, but these are unstable and the motion on the attractor is not cyclic itself.

-- Lou Pecora (my views are my own)

Not to simulate, temperature as the system itself.

I think it is: if the values aren't cyclic then they have to march of to infinity (or the universe ends). They cycle around the 'attractor': a mean value.

-- Nicholas O. Lindan (my views are a product of all the people I met, I don't have a single unique view in me.)

I read in sci.electronics.design that snipped-for-privacy@netscape.net wrote (in ) about 'Neato chaotic equations for analog computers to display?', on Tue, 21 Dec

Intentionally? (;-)

"Lou Pecora" wrote

Anything. A bad way of saying the universe is chaotic, and one can pick any old measurement of it as an example: The temperature goes up and down, staying within bounds and is only predictable in the general sense.

Chaos is safe; order is dangerous. Although I imagine with a .mil address you won't quite agree with that sentiment.

The universe also has a random factor so maybe chaotic isn't the right category of things fuzzy.

wrote

The model may quickly run into precision limits and produce false results.

-- Nicholas O. Lindan, Cleveland, Ohio Consulting Engineer: Electronics; Informatics; Photonics. Remove spaces etc. to reply: n o lindan at net com dot com psst.. want to buy an f-stop timer? nolindan.com/da/fstop/

Not yet. :-)

-- /~ snipped-for-privacy@kltpzyxm.invalid (Charlie Gibbs) / I'm really at ac.dekanfrus if you read it the right way. X Top-posted messages will probably be ignored. See RFC1855. / HTML will DEFINITELY be ignored. Join the ASCII ribbon campaign!

I don't understand that last sentence. Back to the original request: How do you design a circuit to model the temperature of "Anything."

Huh?

Best not to assume what anybody thinks based on their email address.

-- Lou Pecora (my views are my own)

Intentiionally and unintentionally. I understand that SPICE can be a bit 'dangerous' to use in modeling chaotic circuits since some versions (older ones?) use ODE's to model things like transistors and diodes, but other versions (newer?) use lookup tables. The latter, I am told and can believe, will not capture chaotic behavior.

-- Lou Pecora (my views are my own)

run

There are a lot of systems that behavior according to the Shadowing Theorem (see Yorke, et. al and probably others) wherein, roughly, for any calculated trajectory there is a real trajectory that follows it within some epsilon for some specified length of time (arbitrarily long). So models will not 'run well' for a while and then diverge off into 'junk.' So, you're right, you can run the model as long as you like.

-- Lou Pecora (my views are my own)

(snip)

There was a useful thread on sci.fractals in 1997

If you convert the three symmetrical differential equations to an analogous difference equation form, you can get striking chaotic behaviour with a spreadsheet (I used Microsoft Excel)

ODE form

Difference equati hendrik richter says:

And lastly: An even simpler version based on symmetrical differential equation model turns out to be: x(new)= x(old)-(4*sum(old)-y(old)*y(old))/K y(new)= y(old)-(4*sum(old)-z(old)*z(old))/K z(new)= z(old)-(4*sum(old)-x(old)*x(old))/K where sum(old) = x(old)+y(old)+z(old) The value of K can range from 5.5 to 11.5 The following are Excel formulas which can be iterated to produce successive values of the three variables. D2=4*SUM(A2:C2) A3=A2-($D2-B2*B2)/$B$1 B3=B2-($D2-C2*C2)/$B$1 C3=C2-($D2-A2*A2)/$B$1

A three lobed scramble of orbits results. (this quoting myself)

John Bailey

On Wed, 22 Dec 2004 03:02:14 GMT, "Nicholas O. Lindan" wroth:

I have a spice model that demonstrates chaos. The simulation will run as long as you care to let it run and will produce chaotic results that are stable.

Jim