[sdiy] Hopf bifurcation VCOs

[Peter Grenader]

Ian,

Shit...woah...that all sounded brilliant.  Wondering what your Mensa score
was.  By counterpoint - I still feel elated when I screw in a light bulb and
it works. I mean, this is some serious sheet you have going here. Had to
read the first sentence about 20 times, still don't *really* get it.

- P

Ian Fritz wrote:

> Hello folks --
> 
> In nonlinear dynamics a bifurcation is a change in the nature of a
> phase-space orbit as some parameter is varied.  In two dimensions, a Hopf
> bifurcation is a transition from a fixed steady state to a circular
> orbit.  A circular orbit projected onto a pair of axes gives sine and
> cosine oscillations, i.e., it represents a quadrature oscillator.
> 
> A simple prototype for a circular limit cycle derived from a Hopf
> bifurcation is given in polar coordinates as:
> r' = r(1-r^2)   th' = w,
> where primes indicate time derivatives, r is the radial coordinate, th is
> the angular coordinate, and w is the angular frequency.  For r = 1 we see
> that r' is zero, so r remains constant at r = 1, while the angle increases
> linearly in time.  So the system is a spinor with unit length and frequency
> w.  Orbits with larger or smaller r spiral into this r = 1 orbit.
> 
> To proceed with designing an oscillator based on this system, we write the
> equations in rectangular coordinates.  The math is a bit messy, and I have
> simplified it slightly by dropping the nonlinear cross terms.  The system
> is then
> x' = x(1-x^2) + wy
> y' = -wx + y(1-y^2).
> 
> For an electronic-circuit implementation of this system we need two basic
> building blocks.  First we need integrators to relate x and x'  (input =
> x', output = x) and y and y'.  Then we need a pair of nonlinear circuits to
> generate the functions x - x^3 and y-y^3.  It turns out that we do not need
> to generate exactly this function, just a function having the same general
> shape.  This is easily done by adding a linear slope to the response of a
> pair of back-to-back zener diodes.  The circuit requires just the zeners,
> an opamp and four resistors.  It is an important circuit, as it can be used
> in a variety of nonlinear circuit applications including chaos generation
> and waveform folding.
> 
> To build this quadrature oscillator I started by cross-coupling two
> voltage-controlled integrators, similar to what we all use for filters,
> phasors, etc.  These represent the linear terms in the coupled equations.
> Then I added the nonlinear circuit described above around each integrator,
> to implement the nonlinear terms in the equations. This may sound
> complicated, but it only requires two chips: a dual OTA (LM3700) and a quad
> opamp.
> 
> The result is a very nice circuit.  As far as I can tell it always starts
> up without any latchup problems.  The oscillations are not at all sensitive
> to circuit parameters.  In fact the coupling of the nonlinear elements can
> be varied over a wide range, with only a small change in distortion.  I
> recorded the oscillations into Sound Forge and did a spectral
> analysis.  Distortion is very small, with all harmonics  down by 50 dB or
> more.
> 
> An attractive feature of this system is that it can be extended to higher
> orders in the obvious way of coupling several subsystems in a ring.  In a
> slightly different system I am working on for high-order chaos generation,
> I was able to easily make a five-phase oscillator (decature oscillator?)
> with low distortion and non-critical circuit parameters.
> 
> More on that later.  :-)
> 
> Ian
>