[Martin Taylor 961023 13:30]
Bill Powers (961023.0800 MDT)
Martin Taylor 961022 13:30 --
Please, please, please, don't confuse the case of a multi-frequency
oscillator with chaos. They are very different phenomena.
Now I am really confused. I thought that one example of a chaotic system was
a nonlinear oscillator with various degrees of positive feedback.
Yes, that's right, but it doesn't work quite the way you were suggesting.
As the
amount of positive feedback is raised, the oscillator eventually breaks into
chaotic behavior in which no clear frequencies can be see, then goes back to
some repeatable waveform, then becomes chaotic again, and so on until the
region of continuous chaos is reached.
So far, so good.
This describes reasonably accurately one of the "routes to chaos." I'll
detail this route a bit more, below.
I was conjecturing that as the system
is driven further and further into the nonlinear region, the various
harmonics of the fundamental oscillation change in relative amplitude and
strength, sometimes locking together and sometimes drifting in an unrelated
way.
But this isn't the way it works. What happens is not the phase
locking and unlocking of an increasing number of frequencies of oscillation.
In some systems, indeed that can happen, but they are not chaotic systems.
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.
To follow the route to chaos, I'll describe an experiment you can actually
do at home. Set up a tap (faucet) so that it is not quite turned off, and
it drips, say one drip every 30 seconds. Time the inter-drip interval.
If the water pressure is steady, so will be the "IDI".
Now slowly open the tap so that the drip comes faster, say every 3 seconds
(I can't say an actual number, as it will depend on the tap). At some point,
the IDI stops being always the same, and starts having a longer interval
followed by a shorter one, like 2.5 3.5 2.5 3.5... This break starts slowly,
and as you increase the flow, the difference between long and short increases.
You can plot a graph, in which the Nth IDI is plotted on the X axis and
the N+1th is on the Y. So long as the IDI is always constant, you will see
only one point. When you start getting the long-short-long-short sequence,
the point splits into two, which diverge as you increase the flow.
At some point, there starts to be four different IDIs, not two, in a
repetitive cycle: 1.7 4,4 2.1 3.7 1.7 4.4 2.1 3.7 1.7...
Now your graph has four points, each of the earlier two having split into
a pair that diverges as the flow increases.
Keep increasing the flow, and the cycle becomes 8 drips long, then 16,
then 32... Each break point is closer to its predecessor than was the
previous gap, by a ratio that is universal in such systems. The ratio
is called Feigenbaum's number, and it's a bit over 4 (I forget the exact
number and can't be bothered to look it up). Anyway, at some flow not
much faster than led to the period 8 oscillator, the period has become
2^N, where N is larger than any integer you care to name (i.e. it has
passed the infinite threshold:-). The IDI "oscillator" has become chaotic,
and your graph looks like a smear, which resolves into smaller smears if
you look accurately, each of which resolves into even finer smears....
Now increase the flow even more, and at some point you will suddenly
find that the IDI has become periodic again, and stays periodic for
some finite range of flow, before going chaotic again through
a similar period doubling sequence. All this will happen too fast to
measure with a stop watch, but if you had some electronic way of measuring
the drip time, then you could directly display the pattern on your
computer screen.
What's happening physically is non-linear feedback. Just after a drip has
fallen, the remaining water in the tap has a surface that will be bulged
down. As more water comes through, the bulge gets bigger, until there is
a point at which it piches off and the drop falls. The remaining surface
oscillates up and down like the membrane of a drum, while more water
comes to fill up the bulge again. If the oscillation is strong compared
to the thermal motion of the water molecules by the time the bulge is about
ready to pinch off into another drip, then the drop will fall sooner if
the oscillatory acceleration is upward than if it is downward (if it were
downward at 1g, the bulge would _never_ pinch off).
The falling of one drop affects the falling of the next, in a way that
depends on the flow rate. And it turns out that in all such systems
where the nonlinearity is polynomial and at least as strong as a
square law, the route to chaos follows this same pattern of period
doubling at ratios of the criterion equal to the Feigenbaum number.
Things are different if the nonlinearity is exponential (which includes
trigonometric such as sinusoidal). Then you can get what's sometimes
called an "explosion into chaos." The system can go directly from fixed
point behaviour into full-fledged chaos with an infinitesimal change in
the control parameter.
From simple observation, it is often hard to distinguish a multifrequency
osciallator from a chaotic one. Finite-time Fourier analysis necessarily
gives you spectral lines with a finite width, and always has a noise-like
floor. And if the individual frequencies are enharmonic, the period before
the waveform repeats can be very long indeed.
Very simple systems can be chaotic. The complication of the signal does not
necessarily reflect any complication in the mechanism. But in the real
world it takes a complicated mechanism to provide a nice clean periodic
signal (think cesium clock).
Martin