Dan Slater's Article in CMJ, and X to the kth power - how do you do it?

[Sean Costello]

Hi all:

The new Computer Music Journal (Volume 22, Number 2, Summer 1998)
finally hit the stands.  Dan Slater has a fascinating article in it -
"Chaotic Sound Systems."  Y'all gotta track it down.

Anyway, Mr. Slater describes a hypothetical Buchla module (complete with
an "artist's conception" that points out how truly beautiful the Buchla
200 designs were visually) that would implement a Ueda Attractor. As
described, the Ueda attractor circuit is very similar to a
state-variable filter, but with the inverting stage replaced with an x^3
stage (x cubed) and the signal input connected to the first integrator
stage.  Slater's hypothetical module uses an x to the k function, where
the circuit can be converted from a state variable topology to an Ueda
Attractor by changing the value of k (for a state-variable filter, k=1;
for the Ueda attractor, k=3).

How the heck would you do this?  Is there some common circuit that
produces an x to the k function, where k can smoothly vary (i.e. by
non-integer values) between 1 and 4?  x^1 is simple; x^2 could be a
multiplier where the input is multiplied by itself; x^3 could be created
by a second multiplier multiplying the input by the product of an x^2
stage.  But how do you get, say, x^2.74?  And how do you vary these
smoothly?  Could you use feedback from a multiplying stage, where a
signal is multiplied by the output of the multiplier?  Or maybe feedback
in a 2-multiplier configuration?

As always, any and all advice welcome.  I think that this would be a
cool circuit to own. 

Thanks,

Sean Costello