Chaos Theory in Practice: Chua’s Circuit

Chua’s circuit is the simplest electronic circuit that produces chaos—the output of this circuit never repeats the same sequence, and is a truly random signal. If you need a good source of randomness, Chua’s circuit is easy to make and is built around standard components that you might have lying around. [Valentine] wrote a comprehensive guide which walks you through the process of building your own source of chaos.

The chaos of Chua’s circuit is derived from several elements, most importantly a nonlinear negative resistor. Unfortunately for us, this type of resistor doesn’t exist in a discrete form, so we have to model it with several other components. This resistor, also known as Chua’s diode, can be created with an op-amp configured as a negative impedance converter and a couple pairs of diodes and resistors. Other variations such, as the schematic above,22`01 model Chua’s diode using only op-amps and resistors.

The rest of the circuit is quite simple: only two capacitors, an inductor, and a resistor are needed. [Valentine] does note that the circuit is quite sensitive, so you might encounter issues when building it on a breadboard. The circuit is very sensitive to vibration (especially on a breadboard), and good solder connections are essential to a reliable circuit. Be sure to check out the Wikipedia article on Chua’s circuit for a brief overview of the circuit’s functionality and a rabbit trail of information on chaos theory.

Generating truly random sequences

Your brain can’t generate random numbers, and computers can’t either. Most of the ‘random’ numbers we come across in our lives are actually pseudorandom numbers; random enough for their purpose, but ordered enough to throw statistical analyses for a loop. [Giorgio] thought generating random sequences would make for an excellent project, so he whipped up a random sequence generator out of a few Opamps, resistors, and a handful of caps.

[Giorgio] used a Chua Circuit – a circuit that models nonlinear equations – to create a chaotic system. When pairs of points from these systems of equations are plotted on a graph, a fabulous and chaotic ‘double scroll’ pattern (seen above) can be found. After taking oscilloscope probes to different points on his Chua circuit, [Giorgio] watched chaos magically appear on his ‘oscope screen.

The double scroll pattern isn’t exactly random, but since the Z signal of his circuit chaotically varies between positive and negative, the only thing needed to create a random sequence of 1s and 0s is sending the Z signal through a comparator.

After calibrating and sampling his circuit [Giorgio] captured thousands of samples at a rate of 5 samples per second. From a cursory glance, it looks like [Giorgio]’s circuit is at least as good as flipping a coin, but proper tests for randomness require many more samples.

A very, very cool piece of work that is much, much more elegant than getting random bits from a Geiger counter.