[sdiy] VC Chaos

[Ian Fritz]

Hi --

It hasn't been easy to find a way to make a chaos generator suitable for 
modular synths.  Standard circuits (such as the Chua generator) often 
require an inductor, have extremely critical adjustments and do not have an 
obvious means of voltage control.  (Note that I am using chaos in the 
technical sense, not the common sense of just some kind of noise.)

A recent paper in the journal "Chaos" describes a general method based on 
integrator circuits -- basically two integrators and a leaky integrator 
with gain connected in a loop, with an internal coupling path for a 
nonlinear element.  It immediately occured to me that this generator would 
be easy to implement with standard synth VC building blocks, namely two VC 
integrators, a VCA and a VC lo-pass filter.  It was easy to build this 
circuit with four OTAs (two LM13600's) and four opamps (two OPA2227's) and 
to get it working.  I followed the suggestion in the paper of setting the 
time constants of the lo-pass and one of the integrators to be equal (T1). 
The second integrator has time constant T2 and the VCA has gain K.  These 
three parameters are voltage controlled.

I have put photos of some of the observed phase-space trajectories up on my 
website.  For these results, T1 and T2 were kept fixed and K was 
varied.  The first set of photos are plots of one of the output voltages 
against its first derivative.  Starting at the upper left, the first photo 
shows a periodic signal (similar to a quadrature oscillator).  The second 
photo shows the signal after the system has undergone two period 
doublings.  The third photo demonstrates chaotic oscillations.  As the gain 
is further increased, the system exhibits several chaotic regions as well 
as some multiply-periodic fixed orbits.  Finally the system falls back into 
a regular quadrature oscillation.

The second set of figures shows the second derivative signal against the 
first output signal for a number of gain settings.  These are, of course, 
much more complicated and interesting.

Here is the link:

http://home.earthlink.net/~ijfritz/xfer.htm

   Ian