[Martin Taylor 961024 17:15]
Bill Powers (961024.0750 MDT)
What I'm trying to understand is what is physically different between an
oscillator that shows chaotic behavior (constant or in-and-out) and one that
doesn't.
I don't know enough to give a clear distinction. But I can give a rough
approximation. Firstly, let me say that I don't know how to make a
multifrequency enharmonic oscillator, so that doesn't come into it.
Any oscillator needs at least two places to store energy, I think. They might
be the kinetic and potential energy of a ball, or the electric and magnetic
fields in a capacitor and a coil, or something else. This implies that
one can treat the oscillator in terms of the variable of energy stored in
one place or the other and how it changes over time. It could be that,
for example x(t) = a*t + b* (y(t-deltat))^2, y(t) = c*x(t-epst)^2, where
a, b, c are constants.
Those are equations just pulled out of the air, and don't mean anything
in particular.
Now, I don't know the criteria for when (or whether) the circuit described
by that equation would go chaotic, or if it would. But in a sampled system,
it seems that so long as there is a nonlinearity greater than a square
in the feedback loop, there will be parameter values for which the system
goes chaotic, and I would not be surprised if the same criterion applies
to the continuous case as well.
This is clearly an unsatisfactory answer, but I'd have to learn a bit more
to be able to define the criteria more exactly.
I guess another proper statement is that some systems are not sufficiently
nonlinear to go chaotic, no matter what values their parameters are set to,
whereas others will or will not be, depending on their parameters. There
are physical differences between the two kinds of systems, but not between
the chaotic and non-chaotic states of the systems that are affected by
their parameter values (such as gain).
One can't see it in the temporal waveform, but if one did a long-duration
Fourier analysis, the multi-frequency oscillator would show several sharp
peaks with essentially zero energy between the peaks, whereas the chaotic
system that looks much the same in the time-waveform would have a continuum
spectrum, with broad peaks.
Wouldn't that just be a matter of the "Q" (reciprocal of energy losses per
cycle) of the oscillating circuit or system?
I was assuming that the question was of a stable multifrequency oscillator.
Let me dredge up my long-rusted electronic memory, and ask whether it is
not the case that feedback around a filter enhances the apparent Q of the
filter, until at some point the virtual Q becomes infinite and oscillation
commences? I'm wondering whether the inherent Q of the filters in the
oscillator would affect the stability of the oscillator frequency in the
absence of thermal or other noise disturbances.
That's really beside the point you bring up, which is a good one. If the
oscillator is truly unstable in its fundamental frequency, and generates
harmonics...
which would participate in the positive feedback according to their
amplitudes and phases, which would change as the system is driven more or
less into the nonlinearities,...
would this be a chaotic oscillator rather than a multifrequency oscillator?
I'm afraid I don't know the answer to this question. I suspect, without
much strength of belief, that it might well be chaotic, because the
frequency variations of the fundamental would be driven by the feedback
around the nonlinearities. But don't go telling anyone I said so:-)
Your example of the dripping faucet sounds as though it would fit these
criteria.
Perhaps. But what I was trying to illustrate with the drippy tap was the
classical "period doubling" route to chaos. The oscillator keeps doubling
its period at discrete values of the control parameter (water flow). The
succession of values at which doubling happens gets closer and closer by
Feigenbaum's ratio, and at a very precise value of the control parameter,
the period becomes infinite. As the control parameter is increased, the
waveform may change, in the sense that different values of the oscillator
output vary their probability densities, but the period remains infinite,
until at some higher value of the control parameter, the waveform returns
to a short, finite periodicity. With further increases in the control
parameter, the period doubles and redoubles just as before, but quicker.
And so on...
That's the classical route to chaos, and your low-Q non-linear oscillator
might well follow it as you increased the loop gain. I really don't know.
Sorry not to be more precise.
Martin