[sdiy] Logistic Equation chaos engine?

[Grant Richter]

Recursive calculation in the analog domain is bog simple to do, since the
input of the next cycle is the output of the last.

If you use two sample and holds, you need a bi-phase clock. For a single
phase clock, you can do the following:

Take two caps with the solid state equivalent of a DPDT switch, then
commutate between the input an output of any circuit with a JFET follower
input and low impedance output. A CD4053 powered off of +/- 7.5 volts will
work. The new result just "over writes" the old when the cap is connected to
the low impedance output of an op-amp.

This simple algorithm produces an infinite bit serial A/D converter:

V(t) = 2 * V(t-1) if 2 * V(t-1) + -Vref < 0
       else 
V(t) = 2 * V(t-1) + -Vref

Recurse

More correctly, it will attempt to expresses the starting voltage on the
capacitor as an infinite length binary fraction. The output is taken from
the comparator used to test if the result is greater than zero. The signal
is also used to select which of the two computations will be stored in the
"results" cap.

The prototype has run continuously for several hours at 10kHz. It sounds
like white noise of course, and it seems to be working, since the capacitors
do not saturate.

This is taken from Marvin Minsky's "Computation: Finite and Infinite
Machines". Thanks to Mike Murphy for converting Marvin Minsky's equation
into an algorithm.

Other recursive algorithms that look easy to implement in analog are the
Logistics equation and the Fibonacci series. The Fibonacci series is just:

V(t) = V(t-1) + V(t-2) with the initial conditions
V(t-1) = 1 and V(t-2) = 0.

which requires a simple analog shift register. You can patch it up on a
Serge using the existing Analog Shift Register. This could be used to make
envelopes with Fibonacci curves.

Of course, being computed by analog means, the results are subject to
certain limits on accuracy. But since they don't control heart/lung
machines, only make interesting noises, the accuracy limitations may be an
asset!