The Chua Circuit


For decades, musicians have been producing distortion in their instruments by overdriving tubes and transistors in their amplifiers. The amount of distortion could be controlled by adjusting how much the tubes and transistors were over-driven but different kinds of distortion required overdriving different kinds of tubes and transistors. Now, a new way of achieving nonlinear behavior in audio has been developed, allowing the principles of chaos theory, e.g. bifurcation, self-similarity, sensitivity to initial conditions, ljaponov exponents, etc., to be embodied, wielded, and played with in a circuit that uses the deterministic linear gain regions of an amplifier to create a non-linear function. Revolutionary? Maybe... Maybe not. But enough of our yapping... Listen to THIS, THIS, and THIS.

CIRCUIT OUTLINE
Basic Chua Circuit Diagram
L = 18 H (This must have a low DC resistance. See R0 below.
R0 < 6.5 Ohms ( DC resistance of the inductor. How low can you go?)
C1 = 10 nF (no polystyrene!!)
C2 = 100 nF
R = 0 to 2000 Ohm Variable Resistor (Chaotic region around 90%)

Part 1: The Non-Linear Negative Resistor NR
 The core of the Chua Circuit is the Non-Linear Negative Resistor. This is a linear amplifier in the low voltage range, a linear amplifier with a slightly lower gain in the middle voltage range, and a resistor in the high voltage range.
Non-Linear Negative Resistor Plot
This is an easy circuit to build but to test it by obtaining the above plot, you have to do strange things with a function generator, an oscilloscope, and a 1K Ohm load resistor. Unless you have a thorough understanding of function generators, oscilloscopes, and a load resistors, or an uncanny knack for following directions a la circuit diagrams, we would not recommend trying to achieving the above plot from the Negative Resistor. Just go ahead and put in the 4 other components and build the Chua Circuit straight off. The Negative Resistor is so simple, it really difficult to mess it up. If any problems occur in the chua, they are probably not coming from the Negative Resistor, so it is usually not even necessary to test that part of the circuit. Just assume that it works. The Negative Resistor is pretty standard and the chua is more sensitive to the subtleties of the different inductors and capacitors anyways.

Of course, the Negative Resistor IS the heart of the Chua, and there may be much merit in customizing the Chua's behavior by adjusting the different zones of the Negative Resistor. Also, different Op Amps may behave differently and an X-Y Plot of the Negative Resistor may be very useful. These are tasks for the advanced Chua user and should not be attempted by the novice.

Part 2: Pots, Caps, and Coils
 The Chua Circuit is essentially a feedback loop involving the Negative Resistor. An initial voltage within the range of the Negative Resistor's action is mapped to another voltage in its action range and fed back into it again as the new input. Because the Non-Linear Negative Resistor works in such a was that lower voltages are amplified more than higher voltages and voltages higher still are attenuated, an initial voltage, if within the appropriate range, can cycle around the loop indefinitely. Furthermore, the more it goes around the loop, the less predictable its orbit will be. Given a voltage, one may look at the X-Y Plot of the Non-Linear Negative Resistor and determine which amplification range it falls in and predict what the next voltage will be but how many steps into the future and one carry this calculation out to with accuracy. The errors in voltage measurement and Negative Resistor Behavior quickly add up to give unpredictable results.

Now, this loop requires a delicate balance of resistors, capacitors and inductors to achieve. If the impedance of the circuit carrying the Non-Linear Negative Resistor's output back in is too high, than the Negative Resistor, even in its low-voltage/high-amplification zone, will not be able to bring the voltage up high enough, and the circuit will grind to a halt. Alternately, if the impedance is too low, the voltage will loop around the outskirts of the Non-Linear Resistor's Positive Resistive zone, which is not resistive enough to lower the voltage into the more interesting zones.

The problems that we had, building the Chua Circuit were all in the Inductors and Capacitors that we used. For one, the Inductor must have a LOW RESISTANCE. The inductance of an inductor is proportional to the area of the coil. The little ones we first tried to use have many more windings, and therefore, a much higher resistance, than this big thing we found:
18 Henry 6.5 Ohm Inductor
We forgot to put a size gauge in the picture but it is about 6 inches in diameter and around 4 inches tall. Only 6.5 Ohms. Where to find one? Try audiophile places, they might sell them as low-pass filters in cross-overs for like $50-$75. You might be able to scrounge one out of old junky speakers.

You could also try to make one. If you have a few hundred feet of good thick wire lying around, 420 turns of 12 AWG wire around a toilet paper tube should do the trick.

We were also trying to use polystyrene capacitors. No good! It worked fine when we switched to regular capacitors.

Potentiometers are pretty much standard since they don't have any secondary features like caps and coils do. (At least the ones we've come across don't have any noticeable ones - we wouldn't be surprised to find a pot with a capacitance above the Chua's tolerance levels.) We have used both regular old coarse pots and really really fine pots that take 20 turns to sweep through 2Kohm and we would want both on the chua because there are times when you want to sweep through the range quickly and there are times when you want to fine tune the delicately balanced attractor. There is nothing more frustrating than trying to find the frequency at which the Chua bifurcates with a coarse pot because you always overshoot and the attractor goes crazy while you're turning because you are changing the resistance so quickly. However, there is nothing more tedious than turning a miniature pc-board potentiometer with a screwdriver 20 turns to get it within range of the attractor that you want. So, if you have a high tolerance for tedium, use accurate 20 turn pots; if you have a high tolerance for frustration, use course volume control pots, and if you are like me and have a low tolerance of both, use a combination of the cleanest pots you can find.

Part 3: Buffers and Dampers and Amplifiers
 Because the feedback loop in the Chua is so sensitive to impedance, the measurement of the voltage across the two capacitors must be done as non-intrusively as possible. Buffers are great at that. The resistance across the + and - inputs of an Op-Amp is on the order of so many hundreds of mega-ohms that the chua hardly even knows that it's there. Now you can put a potentiometer across the same two capacitors to dampen the attractor cycle at your own discretion. Putting another Op-Amp circuit after the buffer will allow you to amplify or attenuate the two capacitor voltages so that they are of a suitable level for whatever other devices you may wish to feed them into next. Some suggestions:
- Audio Amplifiers and/or Speakers
- Oscilloscopes
- Chua Circuits (Including Itself)
- Reverb/Delay and other FX
- Chua Circuits Through Reverb/Delay/FX

Part 4: Free Running Attractors
 The circuit has five variable resistors. One lies between the two capacitors and one across each capacitor. These three control the behavior of the circuit. Two more control the magnitude of the voltage measured across the two capacitors but don't effect the circuit behavior because they lie behind the buffers. When the circuit is running freely (unperturbed) the attractor that it settles in is determined by the a combination of the three resistances. When the two dampening resistors are at maximum resistance, the resistor between the two capacitors (I'll call it the Separating Resistor) will determine which attractor the circuit runs. Beginning a sweep from the lower side of its resistance spectrum will land the circuit in a low amplitude circular orbit. As the resistance is turned up, the circular orbit will split a little hump off of one side. As the resistance is further increased, the hump will split off into multiple humps that are traversed by the circuit in a non-deterministic manner.
Single Scroll Attractor in Chua's Circuit
As the resistance is further increased, the humped circular orbit will begin to flip vertically in a chaotic manner as well and the flipping will get more and more frequent until a second stable attractor is achieved:
Double Scroll Attractor in Chua's Circuit
The Double Scroll.

Now the circuit can traverse the upper scroll for an indeterminate number of orbits and then switch to the lower scroll of another indeterminate period. Increasing the resistance further will increase the size and complexity of the attractor until the circuit leaves the non-linear region and begins to cycle only around the resistive zones of the Non-Linear Negative Resistor, producing an annoying high pitched tone that is almost impossible to get out of without resetting the circuit.

The dampening resistors do nothing at maximum resistance, but drive the free running circuit towards the lesser attractors as their resistances are lowered. These resistors may have more interesting effects when perturbations are added to the circuit.

Part 5: Perturbation and Dampening
 Well, now that you have your free running chua circuit, what are you going to do with it. Perturb It!!! The initial voltage surge started the attractor by applying a voltage within one of the linear zones of the Non-Linear Negative Resistor. That voltage determined the behavior of the circuit for a few cycles (each cycle taking approximately 1 / 3000 th of a second) but chaos and error and other butterflies from Japan quickly add up to take control of the circuit. Applying an oscillating or pulsed voltage such as from that of a function generator, a musical instrument, or a Chua Circuit*, will reset the circuit with a predominantly predictable behavior many times a second. The frequency and magnitude of the perturbation function and the frequency and magnitude of the Chua attractor can now compete for control of the circuit: The classic struggle of Order vs. Chaos, Good vs. Evil, etc. can now be played with in your very own home. Because the Chua runs at 15 Volts in this implementation, an amplifier has been included to boost microphone level signals to 15 Volts so that they may compete with the Chua. Also, the dampening resistors may be adjusted to give Chua an appropriate handicap. The amplified perturbation can be applied to either of the capacitors for different effects on the Chua. Perturbing across the smaller capacitor seems to lead to a predominantly Chua sound perturbed by the the perturbation signal. Whereas, applying the perturbation across the larger capacitor leads to a sound characteristic of the perturbation, distorted by the chua.

Here are some images of a perturbation of a steadily increasing frequency unwinding the tightly wound coils of the double scroll attractor.
Close-up of Chua's upper scroll Perturbed at 400Hz
400Hz


Close-up of Chua's upper scroll Perturbed at 320Hz
320Hz


Close-up of Chua's upper scroll Perturbed at 260Hz
260Hz


Close-up of Chua's upper scroll Perturbed at 200Hz
200Hz


Close-up of Chua's upper scroll Perturbed at 120Hz
120Hz


Chua Perturbed by Guitar
Open G Chord on Guitar

The Guitar introduces the oval traces because its waves are mostly sinusoidal (triangular, in theory, but whose counting). Inside the oval, however, lies the unmistakable mark of the Chua's Double Scroll Attractor.

Perturbing the Chua with its own output run through a time delay is REALLY REALLY COOL! This is so cool that it can drag the Chua in and out of its heinous High Amplitude Orbit (When it remains exclusively in the Resistive zones of the Non-Linear Negative Resistor), gradually without the need for a reset. With that, we will bring our discussions of the Chua Circuit to an end. Now, go build it and see/hear some of the crazy sounds you can make out of it!



* Perturbing the Chua with a Chua signal is not exactly adding determinism in the same sense that perturbing with a sine wave is. It is, however, constant with respect to the state the Chua circuit may be in at any particular moment and it influences the attractor, accordingly, in a somewhat predictable manner. In other words, because the circuit is locally linear, it is actually a predictable and controllable waveform in its fine grain structure, on the order of a few milliseconds. So operations, such as delayed feedback, on that order can have very interesting effects.