The Rössler attractor challenge.

[GK]

Depending on the enthusiasm and interest of others out there, this thread may either flop or become a catalyst for a veritable flurry of highly exciting activity.

The challenge is to design and breadboard an functioning analog circuit to solve the three coupled differential equations of the Rössler attractor:



Those equations are taken from the Wikipedia page here: http://en.wikipedia.org/wiki/R%C3%B6ssler_attractor

The constants, a, b and c are subject to some variability, but a good place to start is a=0.2, b=0.2 and c=5.7.
The variables can be altered so long as the values chosen still result in a chaotic solution.

Now, being the amazingly clever person that I am, I have already worked out a circuit to do the job. Here it is:



At this stage I won't post the circuit diagram, as that would give most of the game away. However I will reveal that I simplified the initial solution down to six op-amp stages (one quad and one dual package) and one analog multiplier chip. Here is the resultant display:



Ultimately, it would be interesting to see how multiple independent workings of the design problem converge on the same outcome. So, at this stage, here is what I suggest for all those who may wish to take part:

1) Work out and breadboard your circuit
2) Post a photo of the resultant oscilloscope display and give your values for a, b and c.
3) Give your active device count (op-amps, etc) but do not reveal your circuit details or workings.

Only once enough players of the game have posted up their scope display photos as proofs off accomplishment can the circuits and workings be revealed. He or she who solves the problem with the simplest circuit wins!

However, you must build your circuit and demonstrate success with an oscilloscope display photo at a minimum!. SPICE results are not acceptable! There are minor challenges to getting a real life analogue solution up and running that to not present themselves in virtual reality; namely, I can reveal, the issue of scaling so that the solution falls withing the voltage swinging limits of the active circuitry and dealing with offset voltages.

Finally, on the topic of the scope display, unless you have a 3-dimensional projection unit like I have, you will need to build a simple circuit to give the Rössler Attractor, which is a 3 dimensional object represented by three-axes variables, perspective for a 2 axis XY oscilloscope display. Attached immediately below is the simple circuit required. It adds the X and Z solutions/signals in such a way as to give a 45 degree tilt to the projection along the horizontal axis, as shown. You do not have to build this circuit to demonstrate success of course, but your display of the attractor will not be as pretty as it may be otherwise as you will be restricted to viewing/displaying the attractor "end on" only, with a display generated by a selection of only two of the three axis variables at a time.     



Have fun!

[kfitch42]

Speaking of Chaotic electronics, have you ever tried Chua's Circuit? http://en.wikipedia.org/wiki/Chua%27s_circuit

That one has been on my todo list for a while... but given the fact that the inbound flux of my todo list is greater than the outbound flux... and the honeydo list having higher priority...

[Dajgoro]

Could somebody explain what do exactly do this circuits do, and is there any propose for them?

[Paul Price]

Rossler attractor equations are useful in modeling equilibrium in chemical reactions.

[GK]

Speaking of Chaotic electronics, have you ever tried Chua's Circuit? http://en.wikipedia.org/wiki/Chua%27s_circuit

Yep

Could somebody explain what do exactly do this circuits do, and is there any propose for them?

Purdy squiggles on the oscilloscope.

[johnwa]

Well, I have managed to produce some pretty squiggles, though unfortunately I don't think they are the ones we are looking for. There is obviously a bit of clipping going on somewhere, I am not sure if this is anything to do with it.

My circuit sounds fairly similar to yours GK, six op-amps and an AD633.

I guess I will keep working at it, though I probably won't have any time before the weekend. Any thoughts on debugging a circuit like this? I couldn't get my simulation to run at all  .

[GK]

Nice! 

It took me two late evenings to get mine to work, and there was little success on the first night. For getting it to simulate, you often need to introduce a transient of some sort to get things kick started. When simulating stubborn oscillators in LTspice, I often use an ideal current source configured as a pulse generator to inject a brief, small amplitude current pulse into the loop right at the beginning (only) of the simulation. A VCVS configured as a "one shot" pulse generator can also be used, with a series resistor (high value, say 1M to 10M) to inject a "start up" transient current pulse into an op-amp virtual earth node/summing junction or similar.

It does look like you have some clipping there, so the scaling may have to be looked at a little closer. Also your b constant is perhaps too far out of range (hint: consider the contribution of the input/output offset error voltages of the multiplier).
 

[johnwa]

Well, by the highly technical process of connecting up a capacitance substitution box and twiddling the knobs at random, it appears that we now have chaos! Though don't ask me what the coefficients are now  .

On another note, after further research, it appears that it may be possible to reduce the op-amp count. (I found a simple circuit for a different system of DEs, I will post a link later, so as not to spoil the challenge)

[GK]

Great! - besides the clipping issue you've got it.

[GK]

On another note, after further research, it appears that it may be possible to reduce the op-amp count. (I found a simple circuit for a different system of DEs, I will post a link later, so as not to spoil the challenge)

That would definitely be interesting. Unfortunately this thread hasn't been that much of a hit so far! I think I give it another couple of evenings before posting my circuit and SPICE files if there remains to be no further participation.

[Crazy Ape]

Looks like fun, I might give a digital version a go, though other stuff eats all my time at the moment so it won't be for a while.
http://www.eecs.berkeley.edu/~chua/papers/Eguchi99.pdf

More here:
http://iaesjournal.com/online/index.php/TELKOMNIKA/article/download/2503/pdf

[HackedFridgeMagnet]

I would love to try but haven't the time. But don't think your thread is not interesting to others.

[mikeselectricstuff]

Might be interesting to make these produce chaotic audio patterns

[EEVblog]

I want to have a crack at this, but I suspect it'll take a good day of playing at least. Will have to wait until after the Electronex show.

[GK]

Whoa there! I'll hold off on posting any circuit details for a while then!

[johnwa]

Well, I think I have got it down to four op-amps and a multiplier now (after getting some inspiration from the circuit I mentioned previously). Subjectively, it does not seem quite as chaotic as the old circuit, but it still gives quite a good display on the CRO. The clipping is gone too.

Again, I started off calculating the component values to suit the supplied parameters, but I ended up a fair way off these values for best results. I made all the parameters adjustable, and it took a fair bit of 'hill-climbing' to optimise the performance.

I haven't gone too far into calculation of the offset voltages, etc - I just assumed ideal behaviour, and chose the impedances low enough to swamp out the bias currents, but high enough not to cause problems with the op-amps' drive capabilities. Possibly these values could be optimised further. I will post a schematic once everyone else has had a go.

[Dajgoro]

I just remembered that last year I've made a kinda odd experimental circuit with 4000 series logic which included lots of phase shifting interlocked oscillators. I made that as a university project, and it is supposed to be a true random generator implemented as a CMOS IC. The spice simulation of the topology and the 4000 series prototype both gave similar results, so it works. But I've never thought about what would happen if I were to hook the thing to the scope in XY mode. So I now tried it, and the output would just generate a solid green block, but when probing some of the oscillators that are in some way connected I get some very unique squiggles. 

Not really the Rössler attractor, but looks weird, so wanted to post some of the many squiggles.

Sorry for my offtopic post.

[smashedProton]

I am going to try this out when I get back to the lab.   This is probably way outside of my skill level though..

[GK]

Well, I think I have got it down to four op-amps and a multiplier now (after getting some inspiration from the circuit I mentioned previously). Subjectively, it does not seem quite as chaotic as the old circuit, but it still gives quite a good display on the CRO. The clipping is gone too.

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":

[GK]

I just remembered that last year I've made a kinda odd experimental circuit with 4000 series logic which included lots of phase shifting interlocked oscillators. I made that as a university project, and it is supposed to be a true random generator implemented as a CMOS IC. The spice simulation of the topology and the 4000 series prototype both gave similar results, so it works. But I've never thought about what would happen if I were to hook the thing to the scope in XY mode. So I now tried it, and the output would just generate a solid green block, but when probing some of the oscillators that are in some way connected I get some very unique squiggles. 

Not really the Rössler attractor, but looks weird, so wanted to post some of the many squiggles.

Sorry for my offtopic post.

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.

[EEVblog]

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!

[mamalala]

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!

The Atari B/W screens where also black and white, as well as the displays of the first Macintosh computers. A lot of electronic typesetting equipment also used B/W displays.

Greetings,

Chris

[Dajgoro]

I just remembered that last year I've made a kinda odd experimental circuit with 4000 series logic which included lots of phase shifting interlocked oscillators. I made that as a university project, and it is supposed to be a true random generator implemented as a CMOS IC. The spice simulation of the topology and the 4000 series prototype both gave similar results, so it works. But I've never thought about what would happen if I were to hook the thing to the scope in XY mode. So I now tried it, and the output would just generate a solid green block, but when probing some of the oscillators that are in some way connected I get some very unique squiggles. 

Not really the Rössler attractor, but looks weird, so wanted to post some of the many squiggles.

Sorry for my offtopic post.

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.

No, it is not white at all!
Since I don't have a camera I took the pictures with my cellphone, and since it was in the dark, it auto adjusted the white balance so a perfectly good green color became white...

Edit: The screen surface is white, but the trace is green/cyan.

[GK]

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.

[notsob]

I supported TeleVideo equipment for a few years, their ' Paper White ' ascii monitors were very popular when they came out.