[sdiy] Hopf bifurcation VCOs

[Fahl, Romeo]

Is it anything like Joerg Schmitz's Feigenbaum thingy?  The math looks
different... but then again, I'm almost math-blind.

http://www.analog-synth.de/synths/logequ/logequ.htm

 

> -----Original Message-----
> From: owner-synth-diy at dropmix.xs4all.nl 
> [mailto:owner-synth-diy at dropmix.xs4all.nl] On Behalf Of Peter Grenader
> Sent: Thursday, November 10, 2005 3:48 PM
> To: Ian Fritz; synth-diy at dropmix.xs4all.nl
> Subject: Re: [sdiy] Hopf bifurcation VCOs
> 
> 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
> > 
> 
>