The Rössler attractor challenge.

[firehopper]

Does that CRT have a white phosphor, or is that just the way the photos turned out? Never seen a CRO with a white phosphor CRT before.

I remember a short period in the 80's when "paper white" CRT screens were all the rage. None of the green or amber rubbish!

Do you mean monitor CRT's or specifically CRO CRT's? I like the blue phosphor of my Tek 551.

are you sure it was blue, or green with a blue overlay?

[c4757p]

I'd say it is the blue version.

[GK]

P11 phosphor, 460nm, to be specific 

[EEVblog]

Do you mean monitor CRT's or specifically CRO CRT's? I like the blue phosphor of my Tek 551.

Oops, computer CRT monitors.

[johnwa]

OK, good work!

In case anyone out there has an inclination to experiment with 3-D projections, all the information required to get started is here:


The transformation units described were the basis for my "3D Projection unit":

I had a look at your analogue computer thread GK, it seems you are an expert in this sort of thing! A very impressive project.

A few thoughts on the 3D projector: You said you had trouble sourcing the sin/cos pots, and had to resort to a digital solution. Did you consider using analogue multipliers driven by sine function generators - I would have thought this would be more in keeping with the 'analogue' nature of the project!

Another idea - with your RGB CRT display, what about building a second projector, slaved to the first, and offset a few degrees. Then feed the red and blue channels, for a stereoscopic display.

For people interested in visualising the Rossler attractor, the 'flow' hack from XScreensaver provides displays of a number of strange attractors. including the Rossler. Unfortunately, there is no way of controlling which system is displayed - they are chosen at random!

[GK]

A few thoughts on the 3D projector: You said you had trouble sourcing the sin/cos pots, and had to resort to a digital solution. Did you consider using analogue multipliers driven by sine function generators - I would have thought this would be more in keeping with the 'analogue' nature of the project!

Yes, I did! I considered a pair of simplified/scaled-down versions of my sin/cos function module ( https://www.eevblog.com/forum/projects/home-brew-analog-computer-system/msg192626/#msg192626 )  to generate the sine and cosine control signals (for the analogue multipliers) from a linear input provided by a potentiometer. Yes, that would have been all analogue, but a great deal more complex!

However, my analogue computer is to be equipped with a chassis containing 9 of the sine/cosine function modules and two chassis' each containing 10 analog multipliers - so alternatively, I will be able to patch a three (or more) dimensional projective unit as part of a program, should I ever feel inclined to do it all analogue.

That book chapter I posted the first few pages of further goes on to describe stereoscopic displays utilizing a pair of CRT's. It all looks very interesting but I've already have more than enough on my plate for now!
 

[GK]

Well, here is my Rossler Attractor circuit:

http://www.glensstuff.com/rosslerattractor/rossler.htm

[johnwa]

Hi GK,

It looks like your circuit is so similar to mine, that it is barely worth me publishing it! Apologies for the poor image quality. The only real differences are that I rearranged it a bit to minimise the need for inverter stages, and I also made use of the differential inputs on the AD633 (which is cheating a bit, I suppose).

The other circuit that I referred to, for the Lorenz attractor, is at http://frank.harvard.edu/~paulh/misc/lorenz.htm. (Designed by Paul Horowitz, too!) I haven't tried it, though it looks to be much the same idea.

I spoke to Dave at Electronex a couple of weeks ago, he was thinking about a solution without an integrated multiplier chip.  I suppose there are are quite a few ways to do multiplication - logging with diodes and summing, etc, though I would have to look up the specifics of this. It would be interesting to see a few different takes on the idea.

BTW, love the old BWD - there is no place for new-fangled digital rubbish on this thread!

[EEVblog]

I spoke to Dave at Electronex a couple of weeks ago, he was thinking about a solution without an integrated multiplier chip.  I suppose there are are quite a few ways to do multiplication - logging with diodes and summing, etc, though I would have to look up the specifics of this. It would be interesting to see a few different takes on the idea.

Damn, I had forgotten about this. Yes, mine was going to do it without the multiplier, or at least attempt to anyway, I have no idea if it will work until I try it.
I need a spare day, as I suspect it's not a 1 hour work first time job.

[GK]

Hi GK,

It looks like your circuit is so similar to mine, that it is barely worth me publishing it! Apologies for the poor image quality. The only real differences are that I rearranged it a bit to minimise the need for inverter stages, and I also made use of the differential inputs on the AD633 (which is cheating a bit, I suppose).

The other circuit that I referred to, for the Lorenz attractor, is at http://frank.harvard.edu/~paulh/misc/lorenz.htm. (Designed by Paul Horowitz, too!) I haven't tried it, though it looks to be much the same idea.

I spoke to Dave at Electronex a couple of weeks ago, he was thinking about a solution without an integrated multiplier chip.  I suppose there are are quite a few ways to do multiplication - logging with diodes and summing, etc, though I would have to look up the specifics of this. It would be interesting to see a few different takes on the idea.

BTW, love the old BWD - there is no place for new-fangled digital rubbish on this thread!

No, still worth posting! Making use of the differential inputs to the multiplier to insert the "c" coefficient and save an op-amp is now so obvious that I have to slap myself on the forehead for not thinking of that myself. However my excuse is that I originally used an obsolete AD533 (metal can package!) from the junk box, and that part has single-ended inputs only. I upgraded to the AD633 in the published version, as that is a readily available part. Another limiting factor to simplifying the circuit further is that I also required a -x signal for the "perspective" circuit. Unless Dave can come up with a simpler circuit, I guess you're the winner 

The BWD was a freebie; had been sitting in storage at work for well over 10 years until we had a recent clean out and ditched a lot of old junk. It was pretty dirty but it cleaned up well. Still works fine, though the CRT isn't all that bright anymore.

[GK]

The other circuit that I referred to, for the Lorenz attractor, is at http://frank.harvard.edu/~paulh/misc/lorenz.htm. (Designed by Paul Horowitz, too!) I haven't tried it, though it looks to be much the same idea.

OK, knew about that one already - my first chaotic circuit was a Lorenz attractor; essentially an identical circuit to the one Horowitz published, but using two AD533 multipliers and 4 op-amps instead of 3 (an additional op-amp inverting stage was required as the AD533 has single-ended inputs only.

I did a "Rossler attractor challenge" because I wasn't able to dig up any prior art on that one, as far as hardware implementations with analog electronics are concerned.

However I have just dug up this:

http://ncnsd.org/proceedings/proceeding03/html/pdf/325-328.pdf

Bizarrely, the Rossler Attractor isn't referenced at all, though there does seem to be some very strong similarities!

Also, besides the circuit by Horowitz, there appears to have been numerous others in the literature for the Lorenz system. 
See here: http://ccreweb.org/documents/physics/chaos/LorenzCircuit3.html and the numerous references cited.

[GK]

Just for fun, here is my take on a circuit for the Lorenz Attractor. Horowitz cheated a bit in his solution as his 3-op-amp circuit does not solve the y state in the correct polarity, providing its complement, -y, instead. My circuit solves y in the correct polarity, but that takes another op-amp. However this isn't a copy of the circuit by Horowitz, but a rearrangement of the solution. My circuit does not solve and use -y as a working variable to compute x & z, but uses -x to solve x, y & z. I've also used a scaling factor of 0.2 instead of 0.1, for larger signal amplitudes that still fit comfortably within voltage swinging limits.

I've also added provision (jumpers and trimpots) for trimming the output offset voltage of the multipliers. This is necessary as the accuracy of the solution is particularly sensitive to the multiplier output offset voltages and the circuit will not oscillate reliably if they are too great.

The circuit is up and running on breadboard at the moment. I've layed out a PCB for it and will post that (photo + Gerbers) up on a dedicated webpage, along with other details including SPICE simulations files in the near future.

[GK]

Well here it is:

http://www.glensstuff.com/lorenzattractor/lorenz.htm

A circuit description, PCB Gerber files and LTSPICE files and purdy 3-dimensional projections provided. The only thing the page is currently lacking is the introductory photo of the completed PCB connected to the BWD scope (blanked out photo currently being borrowed from my Rossler attractor page).

After doing a little more research (perusing Encyclopedia of nonlinear Science, Alwyn Scott, Editor, Routledge, New York, 2005) I decided to amend the introductory paragraph to the Rossler attractor page:  http://www.glensstuff.com/rosslerattractor/rossler.htm   ...... to properly make the distinction between "Rossler system(s)" and the one commonly referred to as the "Rossler attractor".

[GK]

Just continuing my monologue...........

From the reference here:

http://www.scholarpedia.org/article/Hyperchaos

I've worked out the electrical analog for the hyperchaotic Rössler system having 4 dimensions and recreated the computed results in SPICE. Here it is in simulation:




I've used a scaling (down) factor of 40:1 to get the state solutions to fit the voltage swinging limits of a 10V full scale input multiplier like the AD633.
The "minimal" chaotic Rössler only required a 2:1 ratio. This means that a real world realization of the hyperchaotic analog with +/-15V op-amps and an AD633 will make the output offset error voltage trim of the multiplier 20 times more sensitive than in the 3D case 

Of the 4 dimensions, x,y,z and w, it is the z state that is the pain here. The bipolar x, y and w states fall well within the voltage swinging limits, but the unipolar z state exhibits the largest dynamic range.

I have some ideas on how to mitigate this issue if the basic analog as just presented in the simulation schematic proves to be too critical to trim and thus stabilize the summation of the b coefficient. However it will likely be a few more evenings before I have 4D hyperchaos happening in silicon and displayed on the CRT of the BWD.


........... just for fun I've already made a start on working out the electrical analog for the 9D model presented in that hyperchaos article......... I'm in a masochistic mood tonight.........

[AlfBaz]

Just continuing my monologue...........

Please, think of yourself as a very interesting lecturer with silent but enthusiastic listeners 

[Balaur]

Just continuing my monologue...........

Please, think of yourself as a very interesting lecturer with silent but enthusiastic listeners 

+1

[robrenz]

Just continuing my monologue...........

Please, think of yourself as a very interesting lecturer with silent but enthusiastic listeners 

+2

[c4757p]

Just continuing my monologue...........

Please, think of yourself as a very interesting lecturer with silent but enthusiastic listeners 

+42

[Balaur]

+42

That explains so much!

[GK]

Ha ha.

My 9D simulation/analog study has stalled for now as in that article (again here: http://www.scholarpedia.org/article/Hyperchaos) a value for the parameter ? isn't specified. I will have to try and find the original reference (Reiterer 1998) in the hope that it won't be completely impenetrable The article also makes no mention of initial conditions or sensitivity to initial conditions, which would have been nice.

Also, can anyone here tell me what the : before the = signifies in the "square cell" geometry equations for b1 through b6?


EDIT: Ugh, certain symbols are not supported by the forums font set and a ? has been substituted. The ? I tried to post above is the second character to the right of the = in the first differential equation of the 9D system. Dunno what it is called, looks like a lower case o.

 


 

[c4757p]

The := symbol just means "is defined as".

[baljemmett]

EDIT: Ugh, certain symbols are not supported by the forums font set and a ? has been substituted. The ? I tried to post above is the second character to the right of the = in the first differential equation of the 9D system. Dunno what it is called, looks like a lower case o.

It's a lower case sigma.  (Hey, in a thread like this, that's about as much contribution as I can manage! )

[GK]

The := symbol just means "is defined as".

So in this case, the same as just = ? 

[GK]

Just got my Lorenz attractor PCB loaded and plugged in, though connected to a more respectable oscilloscope this time. New fangled rubbish like that BWD has no place in a thread like this! 

[Stonent]

Just got my Lorenz attractor PCB loaded and plugged in, though connected to a more respectable oscilloscope this time. New fangled rubbish like that BWD has no place in a thread like this! 

I AM AN ANGRY SCOPE! GET OFF MY FRONT END!