woelen
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Mathematics and electronics experiment with chaotic oscillator
This time I did mathematical experiment, researching a very simple non-linear oscillator which exhibits very interesting chaotic behavior and which
also easily can be built and tested with off-the-shelf electronic components (I spent somewhere between EUR 2 and EUR 3 for building it) in half an
hour or so.
I like the experiment from a scientific point of view, it nicely demonstrates some basics of the most simple type of chaotic systems.
http://woelen.homescience.net/science/math/exps/rossler_chao...
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pantone159
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Very nice!
I have been wanting to learn more about electronics and this seems like a nice project.
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stygian
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What does it sound like? Looks like a mobius strip.
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chemoleo
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That's very nice - reminds me a bit of fractal geometry, Julia sets, Mandelbrots and the chaosgame.
Particularly the latter, where with certain parameters, one gets fern or tree like shapes, whilst using the same (much simpler than here) equation but
differnt parameters, one gets lines or random patterns. Same when you play around iwth the Mandelbrot set - simple changes to the equation, and
totally different patterns, sometimes disgustingly bland and alien, at other times quite beautiful.
Your system reminds me of orbits of planets, and particularly moons (i.e. jupiter moons), where at the right mass, angular momentum and effects from
neighbourngin moons, one gets beautiful synchronised motion between moons (i.e. some of the gallilean moons), yet at other times, movement is chaotic
(see the saturn moon Hyperion).
BIt of a ramble - but it's late 
Never Stop to Begin, and Never Begin to Stop...
Tolerance is good. But not with the intolerant! (Wilhelm Busch)
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12AX7
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Gah, styg beat me to it... I wanted to say put a speaker on it!
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chief
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Yeah, nice ...  ... only: What became of the Chaos-science in the last 20 years
? Nothing big turned up ... ; was a nice thing to study when computers became available ...
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franklyn
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I have wondered if quantum indeterminateness is a manifestation of
a chaotic dynamic and therefore deterministic . Discovering the
underlying synchrony would circumvent the exclusion principal.
More of a wishful whimsy than anything else.
Chaotic oscillators are quite a study all their own and has applications
for example in cryptography. Here some related papers
@ pantone159
Here is a good source to search for instructional books
http://openpdf.com/ebook/a-practical-introduction-to-electro...
A Simple Electronic Circuit to Demonstrate Bifurcation and Chaos
http://www.eecs.berkeley.edu/~chua/papers/Hobson96.pdf
Chua’s Circuit – A Universal Paradigm for Generating and Studying Chaos
http://www.oldcitypublishing.com/FullText/JAPEDfulltext/JAPE...
A Universal Circuit for Studying & Generating Chaos Pt I: Routes to Chaos
http://www.ee.berkeley.edu/~chua/papers/Chua93.pdf
http://www.elettrotecnica.unina.it/files/demagistris/didatti...
A Universal Circuit for Studying & Generating Chaos Pt II: Strange Attractors
http://www.eecs.berkeley.edu/~chua/papers/Chua93b.pdf
A Universal Circuit for Studying Chaotic Phenomena
Experimental Chaos from Autonomous Electronic Circuits
Click Request download at Bottom of this page _ http://ifile.it/ftdw23l
and download on the bottom of next page
Attachment: Chaos for encryption.pdf (1.1MB)
This file has been downloaded 946 times
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densest
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Ohhh... way too many years ago I played with this on a real live analog computer - patch panels, huge X-Y display, and all.
I think it was full of 12AU7/ECC82(?) vacuum tubes and the room was always hot.
I think that chaos is in some ways very boring and not very useful in design. I know that some engineering courses teach how to
avoid chaotic behavior because in mechanical systems it often causes failures and in electronic systems causes jumps into operating regimes
which are not desired, especially in things like vehicles and life support systems. I say "boring" because the information content of a chaotic system
is quite low - not zero, but low. As woelen's excellent article shows, there are many apparently different regimes of quasi-stability intermixed with
higher variability ones. Since one part of chaos is "sensitive dependence on initial conditions" by definition it's hard to set up "predictable
chaos". One might figure out how to use it to send really low bandwidth data masquerading as noise, for instance. The area covered by all the
paths (there's an exact technical term which escapes my battered brain) is finite and conveys some information about abstract properties of the
generating system. I think that the ideas of sensitive dependence and hidden order are very useful in analyzing real systems but not necessarily
useful (other than in the negative) in designing systems.
Heh. Franklyn has added a lot of interesting stuff to look at... we overlapped our posts.
One application for chaotic systems in design is generating real-sounding musical instruments. Organ pipes, for instance, are superficially simple but
the chaos in the airflow over the mouth generates a "living" sound which is subtly but unmistakably different from a well-stabilized oscillator, even
one which appears to produce the same waveform. There is really very little more information in the signal but human ears are sensitive to
disturbances in phase and frequency presumably because those signals convey position and velocity information.
I know that many organists can instantly tell whether a series of notes was recorded or synthesized - unless the synthetic notes have a controlled
amount of chaos of the right kind.
[Edited on 25-4-2010 by densest]
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franklyn
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Audiophiles regard vacuum tube amplifiers as the epitome
of sound reproduction, describing it as having an added
' warmth " not found in digital synthesis.
Whether electrical or mechanical many control systems requiring feedback
can be induced to perform erratically ( chaotically ) by out of phase output
fed back into the input. Instead of reinforcing the input this has the effect
of randomly interfering with steady state oscillation.
Early methods of governing steam engines would often result in a condition
called " hunting " in which the speed of operation would not remain steady.
This was caused by a delay in the control feedback - essentially out of phase.
Not related to chaos since this is a well behaved mechanical oscillator, this
remarkable Ignatz swinging pendulum device , a rubber band powered dual
pendulum model is pictured below. Attaching the weights together at the
central axis allows the rubber band to be wound many hundred turns. The
weights are then allowed to swing free as the rubber band unwinds over a
period of some hours. See the videos here for operation
http://www.youtube.com/watch?v=UG0W7_-NVZ8
http://watchismo.blogspot.com/2008/09/ignatz-flying-pendulum...
For more of this Google " ignatz clock "
http://www.google.com/patents?id=pGBTAAAAEBAJ&zoom=4&...
[Edited on 26-4-2010 by franklyn]
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densest
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@franklyn - ooooohhhhh tubes vs. transistors.... tubes give you a big dose of 2nd harmonic which gives the "warm" sound. Some pretty good tube
simulators have been built which do that and make a very similar sound.
The tube vs. transistor hate-fest has a long history. Transistor amplifiers require a different set of design equations and have a different set of
imperfections which have to be worked around. Historically, transistor amplifier designers haven't paid attention to those details which results in
intermodulation distortion especially around the zero point which is the worst place to generate it. IM distortion at a very low level is easy to hear
and grates on the ear. The traditional measure of amplifier quality, harmonic distortion of a single frequency signal, fails to detect the conditions
which generate IM distortion. Engineers building early transistor gear said "It measures 0.003% harmonic distortion, how dare you say it sounds bad"
and the mud slinging has kept on to this day. As an aside, the commonest HP distortion analyzer until about 1980 had a floor at 0.05% so almost
everything measured 0.05% harmonic distortion. When IM distortion analyzers penetrated, there were a lot of red faces.
Many (most?) amplifiers/receiver manufacturers used Sankyo (I think) power modules for many years which are very very cheap, relatively reliable, and
sound pretty bad. So "all audio gear sounds the same" was true, since it was all the same inside despite the label on the case. Now there are very
very cheap "class D" switching power amplifiers which sound horrible in a different way.
Many transistor implementations are easily overloaded by radio frequency waves which have become much more powerful and prevalent. After 30 years,
there are now articles in the electronics trade press about keeping radio waves out of audio gear because a cell phone is a powerful transmitter and
the "blat blat bloooot" of GSM is unmistakable. Even without cell phones, much gear "sounds bad" in an urban RF field.
National Semiconductor, Linear Technology, BurrBrown/TI, Analog Devices, and some other semiconductor makers sell chips with design notes to make
really good audio equipment which is totally neutral - it doesn't have a "sound". That, of course, really upsets the "golden ear"
crowd because it means they have nothing to modify, nothing to rant at, and good equipment can be made at prices mortals can afford. Then they go out
and buy silver cables because the last 6 feet of wire can cancel out the effects of the 1000 feet of wire the signal has been through already.
Recordings also have gone through the analog tape to digital conversion. As with amplifiers, the fiddly little details of making a clean A/D converter
are absolutely necessary and rarely done. Recording engineers (or whoever puts the mikes out and clicks "record" on the laptop) commonly ignore
overload conditions which put garbage on the track. Analog tape overloads gradually. A/D converters go flat on top. In a pinch, a good compressor/gain
follower will avoid horrible distortion while losing some dynamics. For most rock/pop/whatever, that's actually good because the intended audience
listens on a portable MP3 player or in a car where the background noise blots out anything quiet.... and radio play is always heavily compressed to
keep the signal sounding strong.
Loudspeakers are the weakest link in the audio chain. Most speakers distort the signal far more than anything earlier in the chain, insert resonances
which color the sound greatly, overload easily, and deteriorate with time. They can be made well. Mostly they are made to look impressive.
MP3 - to adequately express my contempt would take a treatise. It guts the sound.
I used to build and sell clean sounding audio gear - I did the electronics & my partner did the loudspeakers. We could sell all we could make. It
was a losing battle against the incredible mudslide of pseudo-science, voodoo engineering, and nonsense psychobabble. To this day, there are good
Japanese brands like Rotel which mostly use good design, sound good, and don't cost $30,000 for a 20-watt amplifier (only slightly exaggerated).
[Edited on 26-4-2010 by densest]
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franklyn
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Woelen please excuse the sidestrack from this thread
@ densest
The definitive test at least of dynamic range , first in DVD
http://www.youtube.com/watch?v=0O5aA83GfYc
Purists will swear direct recording to LP cannot be beat
allowing you to hear every snap crackle and pop .
http://www.youtube.com/watch?v=Hb8A-85IamQ
.
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woelen
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I connected the outputs of the oscillator to the line input of a stereo system. The x and y signals sound very similar, they sound like a distorted
sine wave (beep) with some hissing mixed in. Even in the chaotic regions, the central frequency can be heard very well, but it seems to be mixed in
with a lot of noise. In periodic regions you hear a clean beep of the central frequency and you hardly hear the lower harmonics (e.g. a period
doubling would lead to a signal of half the frequency, but apparently that half frequency signal only is weak compared to the central frequency of the
linear oscillator).
I also did an FFT on sampled output and the central frequency always appears as a strong spike in the FFT amplitude plot (absolute value of the
complex vector, given by the FFT software). I used the software from 'Numerical recipes in C', which is very good.
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chief
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The "numerical recipes" are not too good ...
==> Just good enough for the average job ...
Don't forget to test the fft-routines for artifacts, by means of analyzing synthesized signals with known content ... ; signals should be in the same
range of frequency, with similar sampling-rate ...
[Edited on 29-4-2010 by chief]
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woelen
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The numerical recipies are very good in the sense that they supply a general workhorse for a great variety of problems and gets you started quickly.
Of course, specialized software for a particular problem always is better for that particular problem but I found these recipes to be very useful for
many different tasks.
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pantone159
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I have kind of mixed opinions of Numerical Recipes. On the good side, I think their text discussion is usually very good, and they cover a wide range
of topics, so there is a lot of useful stuff.
On the bad side, I find their style of coding to be particularly hard to read. I'd be embarrassed to write code structured as poorly as theirs is.
(However, their style does take a lot less source lines, which matters if you are printing it in a book! My code always uses plenty of lines.)
I am certainly happy to have a copy on my bookshelf.
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densest
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N.R. is a good catalog, I agree.
FFTs are very sensitive to the precision of the floating point representation used. 80-bit FFTs have a noise floor a surprising amount lower than
64-bit, and 128 bit ones get amazingly clean if one is willing to wait long enough for the poor CPU. The choice of windowing method is not obvious -
it was a long time ago that I researched that - but it can affect precision, noise floor, and selectivity. The wrong one can add artifacts/spuriae in
unexpected places or cause signals to be missed entirely.
I always tested my FFTs with a nasty synthetic signal which had a known spectrum by construction. My face was frequently red from embarrassment when
my bugs were revealed.
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bfesser
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Experimental Replication & 'The Symmetry of Chaos'
woelen, I've been thinking about trying to reproduce your experiments with this circuit, now that I have an oscilloscope, but I have
a few concerns that I hope you might be able to help with.
First, I've read that you used the TL072CN in your experiment, but I only have a dozen or so μa741C (vintage Signetics 49th month '73) op-amps
on hand. Would the 741 be a suitable substitute? If not, I'll try ordering some <a
href="http://www.mouser.com/ProductDetail/STMicroelectronics/TL072IN/?qs=sGAEpiMZZMtCHixnSjNA6MwNigq1a7AoHHdGx3%2f4%252be0%3d"
target="_blank">TL072IN</a> <img src="../scipics/_ext.png" /> before they disappear. For the caps, I'll likely need to run to the
local surplus shop.
My other question is what power supply did you use? I'm trying to tell by the rail connections in the breadboard photo, but it's difficult. Does
this require a dual supply, and what voltage do you recommend? Sorry for the novice questions—I only understand a little about the
electronics, and even less of the mathematics, but I find this area fasinating.
Attachment: ua741.pdf (1.1MB)
This file has been downloaded 377 times
Attachment: TL072.pdf (318kB)
This file has been downloaded 367 times
P.S. If anyone can supply a copy of the original Signetics μa741C datasheet, I will love you forever.
<hr width="80%" />
<a href="http://sourceforge.net/projects/symmetrichaos/" target="_blank">Icons</a> <img src="../scipics/_ext.png" /> "Program to
generate the Fractals presented in the book: Symmtery in Chaos by Michael Field and Martin Golubitsky" (SourceForge)
I have a hardcopy of the book (<a href="http://ukcatalogue.oup.com/product/9780195310658.do" target="_blank">The Symmetry of Chaos</a>
<img src="../scipics/_ext.png" /> , and hope to understand it someday. For
now, It's full of beautiful graphs.
[edit] I'm still struggling to get Icons to compile...
[Edited on 5.1.14 by bfesser]
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woelen
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I think any decent opamp will do the job. The uA741 has a much lower gain-bandwidth product (gain*bandwidth is appr. 1 MHz) than the TL071 and TL072
(the latter has two opamps in one package), but at the low frequencies in my experiment (at most a few kHz) I do not expect any problems with the
lower gain-bandwidth product of the uA741.
Both the uA741 and the TL071/72 are frequency compensated and may be used as unity gain voltage followers. The TL071/72 were used, because these were
easiest to get for me. They are very cheap and very common. Similar opamps are TL081/82, they have slightly more noise (but less than the uA741) and
in some locations they are even cheaper than the TL071/72, but where I live, they are harder to obtain. Just ask in your local electronics parts shop.
I used a symmetric power supply, +/- 12 V, but +/- 15 V or +/- 10 V also will do the job.
I would say, any frequency-compensated opamp, suitable as a unity-gain voltage follower will do the job, the circuit is not critical at all.
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bfesser
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Excellent, thank you for the advice, woelen. I'll proceed with the μA741Cs, but I'll put some TL072s on my next Mouser order.
I'll try using a PC power supply for the +/- 12 V. It's too cold outside today (-30°C) to walk there, but I'll make a trip to the surplus
shop soon to pick up the necessary capacitors, and will report back with my results.
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