Chaotica is the name of a multi-featured, autonomous, third-order chaos circuit I have developed. Its features are as follows:
1.) A third order system with a total of four nonlinear elements in the circuit path to provide a wide range of chaotic signals.
2.) Exponential voltage control of rate (frequency) over a wide range, from creeping sub-audio rates to well up into the kHz audio range.
3.) Linear voltage control of loop damping, gain and offset. Many interesting sweep and modulation effects are enabled by these control inputs.
4.) A Reset/Inhibit input which allows the system to be set to zero voltage at all three integration stages. This has many interesting uses. For example, the chaos path's evolution can be followed by varying the rate of a series of reset pulses. By limiting the range of an attractor's evolution, one can produce interesting and complex periodic waveforms.
5.) Voltage outputs for all three system variables, x, y and z.
Below is a pair of composite photos showing a range of patterns that can be produced.
Here is a video demonstrating modulation of various voltage controlled parameters. :
The following sound clips demonstrate operation of the module at audio rate frequencies. The module is in "Double-Eye/Tame" mode. Three LFOs modulate the Gain, Damping and Offset parameters. The "x" output is fed to a Threeler filter, whose parameters are also being modulated.
audio rate demo 1
audio rate demo 2
The next two clips demonstrate audio rate modulation applied to the module, which is itself operating at audio rates. This is somewhat similar to other audio rate modulations schemes, such as ring modulation or audio rate AM and FM, using traditional modules. The mode switches are set for "Single-Eye/Tame" mode. The modulation source is a triangle wave. In the first clip, the Offset input is being dynamically modulated and the Gain control is being increased manually. In the second clip, dynamic modulation is applied to the Damping input, while the Rate input is being randomly stepped, along with the frequency of the modulation source.
audio rate modulation 1
audio rate modulation 2
The remainder of this page is concerned with deriving Poincare sections from Chaotica's outputs and using them for synthesizer control. The oscilloscope photos usually seen for chaotic systems are projections onto a plane. In other words, if you see x-y photos, like those in the first two figures above, it is as if the whole signal path (the chaotic attractor) had been squished flat like a pancake.
Poincare sections, on the other hand, are cross-sectional slices of the attractor, for example a map of all the x-y points where the third signal (z) crosses a fixed value. These are easy to extract using trigger generators and sample-and-hold circuits, such as the TGTSH circuit discussed elsewhere on this site.
The figure below shows an attractor from Chaotica with its z=0 Poincare section -- the dots -- superimposed. There are two sets of dots, one for positive-going crossings and the other for negative-going ones.
was made as an example of electronic sounds created from Poincare sections. For this recording, the z-axis position of the sectional plane was slowly swept across the attractor. Three TGTSH modules were used for acquiring the x- and y- signals defining the section. Then three more TGTSH modules were used to downsample the results to produce the steady tempos heard on the clip. The x and y outputs control the pitch of two synthesizer patches.
The first half of the clip represents the upward crossings through the section for three sweeps across the attractor. The second is the same for the downward crossings.