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Welcome to Calculus. I'm Professor Greist.
And we're about to begin lecture 20, bonus material.
In our main lesson we analyzed the differential equations for a pair of
coupled oscillators and showed that the phase, the angle difference between them,
goes to zero, at least after linearization.
Does this really work? Let's see.
Let's do an experiment. I've set up a pair of metronomes, each
set to Presto, or 184 beats per minute. We're going to couple these oscillators
together by means of a common rolling platform.
That way as each metronome swings from one side to the next, it imparts an
impulse that is communicated to the other oscillator.
Let's set first one, and then the other in motion.
See if the phase difference between decreases, and if so we'll give it a
little perturbation, see if it's stable.
[SOUND]
I think that worked pretty well. Indeed it certainly seemed as though we
saw a wonderful example of a linearized solution, predicting exactly what was
going to happen. But did we really see what we predicted?
Let's take a careful look. If we look at a waveform analysis of the
sound patterns that occurred and we identify where the beats were, then
you'll notice, even after, a long time to let the system stabilize, there's a
little bit of a gap between the two clicks of the metronome.
They didn't synchronize perfectly. It seems to be about six 100ths of a
second. Gap between them.
Now maybe, we didn't let the system run long enough?
No, our prediction says that it's supposed to decay exponentially to a
phase of zero. We shouldn't have such a stable gap.
What might the explanation be? Well we certainly made a number of
assumptions in our model. One of the critical assumptions was that
the frequencies of the two oscillators were identical.
But what if they're not? I had to set the frequencies on those two
metronomes by hand. And it's not likely that they were
exactly the same. Let's rewrite our model to have a pair of
frequencies. The first a1, the second a2.
Then repeating the derivation of the differential equation for the phase.
Phi. What do we get?
Well, we'll get d phi dt equals a2, minus a1, minus 2 epsilon sine phi.
If a1 and a2 are identical, then those terms would vanish as before, and the
previous analysis would hold, with a stable equilibrium at phi equals zero.
However, in this more general case, when we set d phi dt equal to zero, and solve
for the equilibrium, what do we get? We get that phi is arcsine of a2 minus a1
over 2 epsilon. If we, say, approximate that near zero,
the first term in the Taylor series of arcsine would be a2 minus a1 over 2
epsilon. What does that mean?
Well, that means that there's still an equilibrium in this system.
And in fact, there's an equilibrium that is close to zero, but not quite zero, and
depends on the values of a1, a2 and epsilon.
It is still, however, a stable equilibrium.
This slight change in our model predicts something like a synchronization, but for
a phase that is not quite zero. That is exactly what we saw in our
experimental data. Now, do note a few things.
If your coupling strength, epsilon, is very small, if it's a weak coupling.
Or, worse still, if the difference in frequencies is too large, then this arc
sign might not give you an equilibrium at all.
You might fail to have an equilibrium, you might get no phase locking or
synchronization at all. There are so many other interesting
phenomena that we could study by changing parameters.
What happens if instead of two oscillators we have three?
Will they synchronize together? Does it matter how we connect them up, if
we tie them together along a line or if we connect them all to all?
Does the influence patterns change the eventual behavior?
What happens, if instead of two or three oscillators, we consider yet more?
What kinds of features can emerge from this system?
Well, you're going to need something a bit stronger than single variable
calculus to be able to handle thousands of variables.
Now, you may wonder when you're ever going to run across thousands of
metronomes that you need to analyze. But of course, this model applies any
time you have a collection of agents that are linked by some network.
This is going to have applications in Biology, in Neuroscience, in Behavioral
Science, any time you have a network. And indeed, there are so many fascinating
things that emerge when you start looking at behavior of a large number of agents
and the simple things that we’ve done in differential equations just scratch the
surface. For example, it’s possible to have
equilibrium solutions in a network of coupled oscillators that generate waves.
It depends on the topology of the underlying network.
That is, how are things connected up? Loops or large cycles like you see
demonstrated here can give rise to very interesting behavior.
But that's a subject for another course. Perhaps you'll take a course in dynamical
systems at some point and learn how to study such systems.
If you do, you're going to want to know not only single variable calculus, but
multi-variable calculus as well. [BLANK_AUDIO]