Some electronic circuits can oscillate in a “chaotic” way.

When displayed as X-Y plots on an oscilloscope, this then leads to intriguing patterns:

The circuit for the above example resembles a “Colpitts oscillator". It consists of just 6 components and was published by Michael Peter Kennedy in 1994. It was re-published in the Elektor magazine Jul/Aug 1995 issue:

Here is the same circuit on a mini-breadboard (using a BC549C instead of a BC547) and tied to a +5/–5V supply:

The oscilloscope display is actually a 2-D projection of 3 inter-dependent
oscillation variables, and because each cycle is different and the oscilloscope
has been configured with a semi-persistent trace, the result looks like a
mysteriously-fading twisted loop. If it were slowed down, a single dot would be
visible, tracing a path on the screen, but never passing through *exactly* the
same coordinates in the same direction again - hence “chaotic”.

The chaotic behavior appears to come from the fact that the transistor is made to operate in a non-linear region, which changes the resonating properties of the circuit. Minute changes will affect how the oscillation proceeds. The chaotic patterns also change visibly when any component is touched, as this slightly alters the circuit.

Here is the collector voltage with respect to ground (and also the base), the oscillation frequency is ≈ 85 KHz:

Two different traces are visible, illustrating how the oscillation changes from
one cycle to the next. Note how the collector voltage can rise *above* the +5V
supply voltage, due to the inductor.

And here is the emitter voltage, which varies across a smaller range:

Click on the images for a larger view.

For some more circuits, the math, and pretty graphs by the same author, see this paper. There are many other chaotic circuits, but the one shown above seems to be the simplest and is very easy to reproduce.

The GlensStuff.com site has a fascinating range of analog circuits, including a large variety of chaotic ones.

See also “Lorenz system” (Wikipedia) and “Lorenz attractor” (Wolfram) for the 1960’s math behind it all.