SV: Re: SV: [sdiy] QVCO schemo up

[Ian Fritz]

At 11:24 AM 11/12/05, Don Tillman wrote:

>Maybe it should be an actual Hopf Bifurcation device?
>
>Ian, would you say that your approximation of the original equations
>has changed the circuit from being an actual Hopf Bifurcation device
>to a State Variable Filter oscillator?

Well ... interesting question.  First understand that the Hopf bifurcation 
is a *general* type of bifurcation that causes a system to go from a steady 
state at a fixed point (no time dependence) to a stable limit cycle 
oscillation (usually circular, I think).  So the equation pair I started 
with in polar coordinates is just *one* simple example of a Hopf 
bifurcation.  In my system, if I add extra damping then it will  start to 
oscillate as this damping is decreased beyond a certain point.  I believe 
this is still a Hopf bifurcation, although it would take me some work to 
prove it.  (There are general mathematical criteria for the Hopf bifurcation.)

When I first hooked the quad osc system up, I used the building blocks I am 
now using to develop higher order chaos, namely a *damped* integrator 
followed by the zener nonlinear circuit.  This setup showed a Hopf 
bifurcation as I decreased the damping.  After that, I took out the 
integrator damping and so now the circuit always just oscillates. So there 
is no longer a parameter available to actually see the bifurcation occur, 
but I think it is fair to say that the oscillation comes from the Hopf 
bifurcation.  Not particularily useful, just an interesting fact, and a 
tie-in to general nonlinear dynamics theory.


>Why do a Hofp Bifurcation device at all?  Maybe it would be musically
>useful to disturb it from its oscillating state.  Any thoughts?

No reason at all.   :-)   I was learning about the simple prototype example 
and was curious as to what an electronic implementation would be like.  So 
first I converted to cartesian coordinates, then I simplified the system 
equations, then I saw I could build the system with building blocks I 
already had built, sitting on an experimental board.  It only took a couple 
of minutes to hook up.  Then I found that I could leave out one of the two 
nonlinear circuits.  What a way to get a standard, well known result, 
eh?  If I had thought more about what I was doing I might have realized 
that I would end up with a state-variable system, but I just plowed ahead 
because I had the circuit blocks already built.


>So, the State Variable Filter oscillator is well known.  The question
>is, does Ian's variation of the stabilizer with the zeners provide
>better (or otherwise more musical) performance?  I dunno.

Actually my implementation turns out to be almost exactly the same as the 
Burwen circuit in the AD handbook.  I don't think the details of the 
nonlinearity matter much -- all you need is for the (antisymmetric) 
transfer function to cross zero with a negative slope at a finite input level.


>One of the problems with filter-feedback oscillators is that they take
>a while to stabilize.  So, for instance, I would be interested in
>seing how Ian's circuit reacts to a large sudden step change in
>frequency.

Right. We looked at that yesterday.  Both my measurements and Harry's 
simulation show that the system takes a couple of hundred cycles to come to 
equilibrium at power up.  However, once the system is running it seems that 
the frequency can be changed instantaneously.  This is because at the new 
frequency the two signals still have the same phase relation as before, so 
all you need is for the integrators to ramp at a different rate.  This is 
instantaneous.

   Ian