I had some fun this past weekend building an analog computer tointegrate the Lorenz equations. I started with Paul Horowitz'sdesign atand added some frills like a rotary switch to select theintegration capacitor sizes and 10-turn pots and knobs for thes, r, and b parameters that allow you to turn them and see theattractor change in real time as you twist knobs. Lotsa fun.Display is on a 10" X-Y scope so the results are richly displayed.By moving the patch wires you get to view x, y, z, dx/dt, dy/dt, ordz/dt.Are there any other simplistic chaotic systems to try next? Havinga small number of parameters is good (to keep the number of knobsreasonable) and analog multipliers aren't the cheapest thing in theworld so it's nice to keep the number of analog multipliersnecessary small too. (Note that multiplication by a constant isusually handled by a fixed resistor divider and multiplication bya parameter is usually handled by a potentiometer divider. It'sonly terms like "xy" that need true analog multipliers.)Even with the Lorenz equations I think that alternative ways of"seeing"the chaoticity may be useful. For example, right now I'm trying toimagine how to present the results through audio, so that a "leftlobe spiral" might sound different than a "right lobe spiral", andmaybe even modulate it according to the depth in/out of a spiral.(If you haven't seen the equations solved real-time you might notknow what I mean about the spirals: in the x-z plane there are twolobes of semi-periodicity, and on the scope screen at the slowerintegration speeds you see outward spirals from each attractoruntil a switchover point is reached and you end up in the otherlobe.)Any other ideas you guys might have?Tim.

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

Neato chaotic equations for analog computers to display?

Dec 20, 2004 43 Replies

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Nicholas O. Lindan 21 years ago

"real"

Since that time (ancient history, almost a 1/2 day ago) it has been discovered by way of Google that there are (seem to be?) 3 forms of chaotic behavior:

o Can be numerically modeled and the model is always converging to the equation.

o Can be numerically modeled to any length of time if given enough precision.

o Can not be modeled.

The act of the model following (or not) the closed form of the equation is called 'shadowing'.

The name fits, as I am sure 'Only the Shadow Knows'.

Yes, mail me the spice model and I'll get hold of the eval copy of MicroSim. I would like to see it. The only reason to hang around on Usenet is to learn something new (AFAICT the usual lesson is learning you are wrong).

-- 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/

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Brian Inglis 21 years ago

Up/down motion modulates frequency: say 5kHz centre with +/-4kHz range, maybe log scale; left/right motion pans the stereo location.

Thanks. Take care, Brian Inglis Calgary, Alberta, Canada Brian.Inglis@CSi.com (Brian[dot]Inglis{at}SystematicSW[dot]ab[dot]ca) fake address use address above to reply

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shoppa 21 years ago

Wow, your hearing range goes really high if you can hear that. Put it down

3 octaves, then maybe it'll work for me :-).

For the traditionally chosen Lorenz parameters (s=10, r=28, b=8/3) the separation between the two lobes is complete enough and so nicely divided by the x-axis that this ought to work well.

Tim.

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Brian Inglis 21 years ago

That I can hear it doesn't mean I'll like it. You can't hear 1kHz-9kHz, but you can easily distinguish ~100-1900Hz?

Figure it might sound like a really high speed Theremin.

Thanks. Take care, Brian Inglis Calgary, Alberta, Canada Brian.Inglis@CSi.com (Brian[dot]Inglis{at}SystematicSW[dot]ab[dot]ca) fake address use address above to reply

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