[sdiy] New Kind of Chaos Circuit

[Oren Leavitt]

On 2/24/2011 6:10 PM, Ian Fritz wrote:
> 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


Sounds very interesting Ian!
I shall stay tuned...

- Oren