Chaos theory is the study of the behavior of dynamical systems that are highly sensitive to initial conditions. Some examples of chaotic systems are : the weather, double pendulums, and Conway’s game of life.
Another example of a chaotic system that you may not have heard of is the Lorenz Attractor. This is a system of ordinary differential equations that are special because they have chaotic solutions for certain initial conditions. That and the fact that when you plot the output it looks kinda like a butterfly’s wings.
It turns out you can solve ordinary differential equations using simple op-amp circuits. Paul Horowitz, author of the famous book The Art Of Electronics took an interest in the Lorenz Attractor and made a circuit to solve the Lorenz Attractor equations. The circuit he came up with looked like this:
That’s about as old school cool as it gets really. The circuit produces three separate voltages, x(t), y(t) and z(t). If you hook up the x and z into an oscilloscope, you get the characteristic butterfly wing plot that looks something like this:
If you want to experiment with this circuit yourself, Hackaday.io user Tom Quartararo has recreated Paul Horowitz’s circuit created and made it into a lovely PCB with some tweaks added in to make it easier to get up and running. For more information on the circuit designed by Horowitz look here.