A main goal in developing chaos circuits for electronic music applications was to develop circuitry that would allow the "frequency", or rate along the chaotic orbit, to be varied over a wide range, as is the norm in synthesizer VCOs and VCFs. This may be done by basing the circuitry on integrators with voltage controlled integration rates, similarly to what is done in synthesizer VCFs.

A high-performance variable-rate integrator can be built using an operational transconductance amplifier (OTA) followed by an opamp integrator. What remains to be explained is how to relate this response to the mathematical differential equation set and the requisite nonlinear circuit element. The following figures illustrates a typical example of how this may be done.

The first figure illustrates a voltage-controlled damped integrator, which may be used to implement one of the coupled first-order differential equations of a chaotic system. The OTA multiplies its (small) differential input voltage by a factor proportional to its control current to produce an output current I_o. Characterizing the OTA as an effective resistance R_OTA yields Eqn. 1. Equation 2 describes the response of the opamp integrator, whose output is a voltage ramp with rate proportional to I_o and thus to the control current I_c. Combining the first two equations results in Eqn. 3.

In Eqn. 3, note that time occurs as a single factor on the left side. Thus by rescaling the time variable, we may cancel the two prefactors on the right side. Note that the form of the scaled equation is then independent of the OTA control current. This means that the shape of the attractor is independent of the rate (or "frequency") of the system. Rewriting the result in a more mathematical-looking manner results in the boxed equation. Here, the time derivative of x is equal to a damping (negative feedback) term plus a term coupling x to a second variable y. Coupling to other variable can be done by providing additional input resistors connected to the (+) and (-) OTA inputs.

The boxed equation may be represented by the highly schematic drawing in the lower box, from which the correspondence to an actual circuit is easily seen.

For a representative nonlinear circuit element, we will use the circuit in the following figure. The transfer function of this circuit (after appropriate scaling of the voltage variable) is similar in shape to the function x(1 - x²). This function is of interest because of its occurrence in many physical systems, e. g., those with double well potentials. The response curve of this circuit is smooth and rounded because of the soft knee of the zener-diode response.

An Example System

As an example of the above methods consider the system illustrated in the next figure. The block diagram follows immediately from the system of equations. From the diagram we see that the circuitry needed consists of three integrators with identical damping connected in a loop with a nonlinear element of the kind described above plus an additional cross-connection represented by the 1/a signal path.

This system has been patched up using the circuitry described in the "Schematics" section. The scope shots below show the x-z and y-z projections of the chaotic attractor for a particular combination of damping and coupling parameters. Below the photos are the results of a simulation carried out by solving the system differential equations using the Mathematica programming language. Values of the parameters in the simulation are a = 0.4, b = 0.4956 and c = 1.1. The agreement with the observed attractor is remarkably good, given that circuit nonidealities can significantly modify the system behavior and that the nonlinear response is only approximately correct.

Here is a brief demo sound clip of this attractor controlling a synthesizer patch.