Here is a photo of a chaotic attractor produced by a circuit incorporating three electronic integrators and two step-function nonlinearities. Each step function is responsible for a double scroll in the attractor. An accompanying sound clip was produced by a patch where the chaos circuit controlled several synthesizer parameters, while itself being repetitively pulsed as the integration rate was varied in steps from low to high and back.

Background

What exactly is chaos, anyway? As with many terms, its technical meaning is different from the meaning in everyday usage. Broadly speaking, a chaotic system is one where the system's variable quantities satisfy deterministic mathematical equations (i. e., present values may be calculated from past ones), while at the same time being highly irregular, but still contained within a finite region (the "strange attractor").

In the system used for the above photo, the variables are three time-varying voltages in an electronic circuit. Two of these are shown in the photo, plotted parametrically (i.e., voltage 1 vs. voltage 2 as time is varied.) In a deterministic system, the path followed by the variables cannot cross itself, therefore a chaotic system must be at least three dimensional (third order or higher). As a special case, one of the variables may be a time-dependent driving force or signal. Thus, where the traces seem to meet and cross in the above photo, they are actually crossing one above the other, utilizing the third dimension (voltage 3).

Electronic Implementation

In making an electronic circuit exhibiting chaos, one is essentially making an analog computer for the differential equations describing the system. This means that circuits that can differentiate a voltage are needed. In practice, it is convenient to actually do this with electronic integrators, the idea being that the input of an integrator is the derivative of its output. In addition, a nonlinear circuit element must be incorporated, as linear systems cannot be chaotic.

Many chaotic electronic circuits have been developed, and chaos was even observed in some the earliest work on vacuum-tube electronics. Most of the studies on chaotic circuits have been aimed at developing simple circuitry. In the present work we emphasize more complicated chaotic systems with variable patterns and evolution rates, which allow a wide variety of resources for potential musical applications.

Variable-rate integrators can be built using operational transconductance amplifiers (OTAs), in the same way that OTAs are used in variable-frequency filters for electronic-music applications. Simple nonlinearities using the response of inexpensive components such as diodes, zener diodes and opamp comparitors can produce useful results, avoiding the necessity of complicated break-point or electronic multiplier circuits.

Reference

J. C. Sprott, Chaos and Time-Series Analysis, OUP (2003). (Includes several circuit examples)

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