[sdiy] New Kind of Chaos Circuit

[Ian Fritz]

Hello --

There are two types analog circuit realizations of chaotic systems, at 
least as far as I have seen.  First, there are "autonomous" 
(self-oscillating) circuits, which require (at least) three 
integrators.  Second, there are driven circuits, which require only two 
integrators, but which have to be driven by an external excitation signal, 
usually a Sin(wt) function.

But there is another important type of chaos system, namely systems with 
two integrators driven by a traveling-wave signal, ie a Sin(kx - Wt) 
function, representing a plane wave wave traveling with velocity W/k. This 
is an area of chaos theory which has been important historically, as it 
provides models for E-M waves interacting with plasmas, etc.

So how do we do this in a circuit?  It turns out to be pretty simple if you 
remember your FM theory and how through-zero FM VCOs  operate.  The circuit 
is based on two opamp integrators connected in tandem, the second deriving 
x' from x and the second deriving x'' from x'.  The circuit is completed by 
cross coupling the integrators and adding any needed linear and nonlinear 
components to complete the realization of the appropriate differential 
equations.

To get the driving signal we use a bit of a trick.  The phase of the 
driving signal is to be kx - Wt.  From this, it is clear that the 
instantaneous frequency needs to be kx' - W.  Already we have seen that x' 
will be generated in the main part of the circuit. Multiplying this by a 
constant and adding another constant (k and W), we easily generate the 
instantaneous frequency.  To obtain the Sin(kx - Wt) drive signal, then, we 
just have to drive the linear FM input of a VCO with the instantaneous 
frequency kx' - W. The output of the VCO is then (almost magically) the 
desired Sin(kx - Wt) signal that drives the main chaos circuit.

So I white-boarded the main circuit of a second-order system and connected 
it to a TZ FM VCO.  This took about 20 minutes.  After about 2 hours of 
messing around I ended up with a working circuit.

The system has a lot of control parameters, but it also produces an amazing 
variery of interesting chaotic signals (along with the corresponding 
non-chaotic limit cycles, of course).

I'll be trying to put up some scope shots and a schematic soon.  Stay tuned!

Ian