It was thought that a two-dimensional version of the simple iterating circuit should be investigated. Simulations are a little more difficult to think about so it was decided to conduct experiments. The circuits were designed and constructed from operational amplifiers, having differing adjustable gains, and with differing output ranges before saturation. In each of the two channels, one for each dimension, there were three saturating operational amplifers whose outputs were combined in a summing amplifier. Control of the gains and the sizes of the Features was by ten-turn accurate potentiometers with precision dials to allow parameters to be recorded and reset. It was arranged to have offset controls in each channel for the position of the Features.
Two two-stage sample-and-hold circuits were used to transfer the outputs of the transfer function generator back to the input, on clock pulses provided by an external generator. A sum and difference rotation matrix circuit was used to mix the two channels thus generated, for the 2-dimensional experiments reported below. In order to keep the range of the input voltage and the range of the output voltage commensurate, the matrix circuits add and subtract 0.5 of the ouputs of the individual channels. For a true ``area preserving" matrix we would require a rotation such that 0.707 of the outputs were taken to add and subtract. Our matrix therefore rotates and contracts, by an area amount of a factor 2. However, since the individual gains of the transfer function circuits are close to magnitude 3, a small area expands by a factor 9 (3 in x times 3 in y) on passing through the transfer function part of the system so the loop area gain is 9/2. Thus we can use this method to construct a chaotic system of arbitrarily large dimension. This is best implemented by using the transfer function circuit sequentially. In a 4-dimensional system one would store a 4-vector at input and output in individual 4-wide transfer and hold circuits.