Chaos theory is an area of study that looks at dynamical systems which exhibit behaviour that is highly sensitive to initial conditions. This sensitivity means that these systems, while not truly random, are impossible to predict. They often exhibit weird patterns of behaviour like fractals, loops and near repetitions.
These are deterministic systems in that sense that their future state can be predicted from their current state – it just that the future state vary wildly if the current state is only very slightly different. Lorenz described these systems thus: “Chaos: When the present determines the future, but the approximate present does not approximately determine the future”.
Electronic circuits can exhibit chaotic behaviour. Chua’s circuit is one of the simplest examples. It is a nonperiodic oscillator that is pretty easy to build or simulate. It was first described by Leon Chua in 1983 while he was a visiting academic at Waseda University in Japan.
The circuit is shown in Figure 1. It consists of two capacitors, an inductor, a resistor and a special element known as a “Chua diode”. This is an active non-linear negative resistor with a characteristic shown to the right of the figure. You can see this has a negative resistance since the slope of the V-I curve is negative. It is also obviously non-linear due to the sudden changes in slope.
Chua’s circuit uses the minimum number of components to create chaotic behaviour. The resistor, capacitors and inductors are all bog standard. The “Chua diode” is an active non-linear device with the negative resistance characteristic shown at right. We have to build this ourselves.
It is active since it operates in the 2nd and 4th quadrants where the product of V and I is negative. This means it “dissipates” negative power – or put more simply it exports power to the circuit.
Of course, you can’t buy a Chua diode from DigiKey – you have to make one. Figure 2 shows two different ways to do this. Both use op-amp based negative impedance converter circuits. The circuit on the left, ignoring for a moment the two diode-resistor networks, is a negative impedance converter. The resistance between the terminals Rn is given by the equation Rn = –Ra(Rc/Rb). If Rb is equal to Rc, then Rn = –Ra. I covered negative impedance converters in an earlier Quick Bits article if you are interested.
Here are two circuits which can emulate a Chua diode. Both make use of negative impedance converters (one is used on the left and two on the right).
So the circuit on the left has a negative resistance Rn in parallel with the two diode networks. If the input voltage is small, the diodes are not forward biased, and we just see Rn. If the input voltage increases in either polarity to the point that one of the diodes conduct, Rx is effectively placed in parallel with Rn. As long as Rn is smaller in magnitude than Rx, the resulting resistance will still be negative, but its magnitude higher than Rn (I know its counterintuitive, but you can check the maths for yourself). This yields something like the curve shown in Figure 1.
The circuit on the right uses two negative impedance converters in parallel. This time the gain of the converters are different, so they provide different impedances. One is set up to enter saturation at a lower voltage than the other, producing the break in in slope. This circuit was proposed in a paper by Michael Kennedy which gives a full explanation of how it works.
All this theory is interesting, but I wanted to check it out in simulation. First, I just simulated the Chua Diode with the values proposed by Kennedy in his paper. The plot in Figure 3 shows the V-I curve as the voltage ramps from -5V to +5V. You can clearly see the discontinuity at in the current at around ±1V.
This is the V-I characteristic of a simulation of the right-hand circuit of Figure 2, using component values taken from a paper by Kennedy. You can see the change in slope of the curve at about ±1V.
With that working I added the rest of the circuit as shown in Figure 4. I used jellybean LM358 op amps which seemed to work just fine. I ran the simulation and plotted the voltages at the nodes labelled V1 and V2 with the results shown in Figure 5. The plot on the left is an X-Y plot of the two voltages clearly showing the features of a classic double scroll chaotic attractor. The voltages V1 and V2 are plotted against time on the right where you can see the aperiodic nature of the waveforms. The simulation is very sensitive to the value of resistor R, and you can get a variety of interesting results by tweaking it.
This is the full circuit as simulated. It is pretty simple but produces some very unusual results. We are interested in the voltages at the nodes marked V1 and V2. Changing the value of R1 has a huge effect on the output.
The voltages at Nodes V1 and V2 of the circuit shown in Figure 4 are plotted on a X-Y plot on the left and against time on the right. The non-periodic relationship produces a classic double-scroll attractor.
I have not built this circuit for real yet, but it would be easy to do. I would probably use a gyrator, built from two op amps, in the place of the inductor and a trimpot for the resistor R. The whole circuit would therefore just need a quad op amp package and handful of resistors and capacitors. Might be a fun weekend project!
Bibliography
“Chua’s Circuit.” In Wikipedia, June 18, 2024. https://en.wikipedia.org/w/index.php?title=Chua%27s_circuit&oldid=1229738530.
“(PDF) Robust OP Amp Realization of Chua’s Circuit.” Accessed August 8, 2024. https://www.researchgate.net/publication/2298819_Robust_OP_Amp_Realization_of_Chua’s_Circuit.
Sponsor this Article
Andrew Levido (andrew.levido@gmail.com) earned a bachelor’s degree in Electrical Engineering in Sydney, Australia, in 1986. He worked for several years in R&D for power electronics and telecommunication companies before moving into management roles. Andrew has maintained a hands-on interest in electronics, particularly embedded systems, power electronics, and control theory in his free time. Over the years he has written a number of articles for various electronics publications and occasionally provides consulting services as time allows.
