[sdiy] Hopf bifurcation VCOs

[Ian Fritz]

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