Motivated by the article, __How to Generate Chaos at Home__
by Douglas Smith in the “Amateur Scientist” column of
*Scientific American* (Jan 1992), Dan Blackwell
investigated the behavior of chaotic circuits. Chaos is the seemingly
random behavior of a deterministic system. A deterministic system is
one where the future state of the system is completely determined by
the current state of the system, while the future behavior of a
random system bears no correlation to its current state. A chaotic
system is deterministic, yet unpredictable in the sense that even a
small uncertainty in the current state of the system leads rapidly to
unpredictable behavior.

Dan developed the following circuit which demonstrated chaotic behavior.

On a conceptual level, the circuit consists of four components: a 20 Ω resistor, a 1.0μF mylar capacitor, a 1N5230 4.7 volt zener diode, and a 100 Henry “virtual inductor” built from a network of resistors, a 1.0μF mylar capacitor, and two LF411 op amps. Thus, the circuit can be seen as a nonlinear LRC circuit, with the diode acting as the nonlinear element. This circuit starts to show deviations from linear behavior when driven at about 130 Hz. The output can be digitized and analyzed by computer, displayed on an oscilloscope, or connected to an amplified speaker.

The output voltage in the circuit is shown in the plot below when the circuit is driven at 1 kHz with an amplitude of about 3 volts. The output shown is irregular, but periodic: it is not yet chaotic.

The voltage output can also be analyzed using a phase plot showing the voltage at one time t plotted against the voltage at time t-1

For a simple sinewave output (when the circuit is driven at about ½ volt), the phase plot is a simple ellipse. As the output becomes more chaotic, the phase plot displays more complex behavior. For example, the curve doubles back on itself near the point (-2, -2) in the above plot: this doubling back gives rise to the phenomenon known as period doubling. A large succession of period doublings leads to chaos.

Some useful links:

This site explains nonlinearity and chaos:

http://amath.colorado.edu/faculty/jdm/faq-%5B2%5D.html

This site references another article in *Scientific American *by
Joseph Neff and Thomas L. Carroll describing how to make chaotic
circuits that can be synchronized.

http://www.fortunecity.com/emachines/e11/86/circsync.html

The site above also presents a nice introduction to chaos:

http://www.fortunecity.com/emachines/e11/86/mastring.html

This site lists several types of chaotic circuits:

http://chaos-mac.nrl.navy.mil/circuits/chaotic_circuits.html