[From Bill Powers (960711.0815 MDT)]
Bruce Abbott (960710.2045 EST) --
Synchrony of free-running
oscilllators can be achieved simply by perturbing them at intervals
close to the free-running frequency, like pushing on a pendulum at
nearly the same point in each cycle.
And how does "perturbing them at intervals" bring them into
synchrony? I don't follow. I can see how that would alter their
periods (temporarily), but why would they synchronize as a result?
First consider phase. Suppose you're perturbing the swing at exactly its
natural frequency, but applying small force impulses to it 90 degrees
out of phase with the swinging (leading). This will cause a small phase
advance every time you push. When you stop applying the push, the phase
of the swinging will now be advanced relative to an absolute time scale,
and will stay advanced since there is nothing to change it (sort of like
Newton's first law).
Now suppose you have two swings side by side, with weights on them. The
ropes are of the same length, so the natural periods of the swings are
the same. Connecting the swings is a weak rubber band. If you start the
swings going 90 degrees out of phase, the rubber band will alternatively
stretch and relax as the swings get ahead of or behind each other. Each
little pull when the swings move apart tends to slow one swing and speed
up the other. The phases will gradually drift, until the two swings are
moving exactly in synchronism and the rubber band is always at its
minimum stretch, pulling neither swing ahead or behind.
If one weight is very much more massive than the other, the periods of
the swings will still be the same, but now the pull of the rubber band
affects the lighter swing much more than the heavier one. The lighter
swing will have its phase changed far more than the heavier one; it will
come into synchronism with the heavier one without much change in the
phase of the heavier one's swinging. This approximates the case of a
one-way effect.
Finally, suppose the heavier weight is on a swing with somewhat shorter
ropes than on the other swing. Now the "driving" swing, the heavy one,
will have a natural frequency somewhat higher than that of the light
one. The phase of the lighter swing will drift until the average effect
of the rubber band becomes constant on every cycle. And it will CONTINUE
TO DRIFT, because the light swing is always being made to move ahead of
where it naturally wants to be. A constant rate of change of phase
(relative to an absolute time scale) is exactly equivalent to a change
in frequency: the lighter swing is now being forced to oscillate
somewhat faster than its natural frequency. It is being forced into
synchronism with the heavy swing, both in phase and in frequency. Of
course it will always lag in phase a little bit, because that lag is
what stretches the rubber band and applies the force that keeps the
lighter swing's phase advancing -- or if you like, keeps the lighter
swing's frequency higher than the natural frequency. But the two swings
will be locked together in frequency. The greater the difference in
natural frequencies, the greater will be the final phase difference.
A swing has no internal source of energy to keep it going; eventually
both swings would run down and stop. But suppose that each swing was
instead a clock pendulum. Each clock maintains the amplitude of its
swing through an escapement mechanism, a simple feedback control system
that draws on the energy of a falling weight or a spring to maintain the
swinging of the pendulum against frictional losses. This feedback system
is relatively easy to perturb, since the force applied in either
direction by the escapement is very small. Basically, all it does is
keep the amplitude of the swing constant. If the swing is too small, the
escapement exerts an aiding force; if too large, an opposing force. So
the escapement is a little control system all in itself, powered by the
weight or spring. The frequency of the clock is determined by the length
of the pendulum.
If you put two closely-regulated pendulum clocks on the same shelf,
eventually they will be ticking in synchronism (with some phase lag, but
at exactly the same frequency). They will keep exactly the same time
from then on, so (as astronomers discovered long ago) the two (or three
or four) clocks can't be used to check each other. What brings them into
synchronism is the microscopic deflection of the shelf as the pendulums
swing back and forth. This is just like the rubber band connecting the
two swings, except that it is an exceedingly weak rubber band.
However, the transmitted force required to achieve synchrony can be
minute because there is no natural stabilization of the phase of a free-
running oscillator. The size of the synchronizing perturbations simply
determines how long it takes to achieve synchrony. The slightest timed
perturbation will gradually shift the phase of the pendulums without
limit. For the clocks, this perturbation ceases (or becomes constant)
only when the two clocks are running at the same frequency, so the
average rate of change of phase becomes constant and equal for both
clocks: one clock's phase advances relative to its unperturbed phase as
the other retards at the same rate.
These are the principles behind all phenomena of biological
oscillations. Circadian rhythms are an example. If bodily processes have
natural oscillations with periods near one day, small perturbations of
any kind that occur once per day will tend to drag the frequency of the
oscillations toward 24 hours (or whatever the timing of the driving
events is). There might be a final phase difference between the driving
events and the natural frequency of the circadian cycle, but the
frequency will be the same if the perturbation is large enough to cause
the phase of the natural oscillator to advance or retard enough on each
cycle.
The larger the effect of the timed perturbations, the more rapidly the
phase of the natural oscillators will advance or retard, and the greater
the frequency difference can be between the driving events and the
natural frequency of oscillation (the "free-running" frequency).
Remember that there is no "restoring force" for the phase of an
oscillator; if perturbed briefly, the oscillator will simply start
oscillating later or earlier, and continue with that new phase until
perturbed again.
A natural oscillator would be some sequence of processes, each one
leading to the next, in a closed circle. Such a process would have a
natural periodicity, not controlled with respect to any particular
frequency but simply occurring at the frequency set by the speed of all
the individual processes in the circle. This process would not be under
feedback control; if anything changed in one or more of the processes,
the frequency would change (which is exactly equivalent to starting a
sudden continuous shift in phase relative to the old phase).
However, it is possible for one natural oscillator to come under the
control of an active control system. One way is to use a reference
oscillator, like the quartz crystal in a wristwatch, except that the
crystal might be used as a reference against which to alter the
frequency of another oscillator, like a pendulum. With active control,
the larger oscillator would have some element in its circle of events
that could be altered in speed or timing of operation so as to keep the
larger oscillator in step with the reference oscillator. If the
reference oscillator is shielded from perturbations, the main oscillator
will suddenly prove very resistant to synchronization with external
events, even strong and carefully timed events. There are other ways to
control frequency that depend on a sharply-tuned perceptual function
rather than a reference oscillator, but that is for another time.
The point is that if the frequency of a natural oscillator is actually
under control, there must be some reference against which its phase,
frequency, and amplitude are compared, and some active way of altering
these aspects of the natural oscillations to keep them from changing. If
external perturbations then tend to alter the phase, frequency, or
amplitude of the natural oscillator, the control system will oppose
their effects and keep the oscillator's behavior from being altered.
So if we find that a natural oscillatory phenomenon such as the one
involved in the menstrual cycle tends to come into synchronism with
other similar natural oscillators, we can be fairly sure of two things:
1) there is some form of perturbation of one oscillator by another, and
2) the oscillation that changes phase or frequency is not under feedback
control relative to any fixed reference frequency or phase.
Of course this does not rule out control systems with variable reference
frequencies or phases. It is possible that either or both of the
independent systems senses lack of synchronization and deliberately
alters its own internal reference so as to achieve synchronization. This
would make synchronization a goal of one or both systems, and not simply
a natural passive outcome of physical interactions.
If one understands the properties of free-running oscillators and their
interactions, it is possible to devise experiments to measure the
properties of each oscillator alone and several that interact. I don't
know how sophisticated the analyses done so far are.
···
-----------------------------------------------------------------------
Best,
Bill P.