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 them 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
one hundredths of a second. Gap between them. Now maybe we didn't let
the system run long enough. No our prediction says that it's suppose
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. But 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- a1- 2 epsilon sin phi. If a1 and a2 were 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 arcsin(a2- a1. Over two epsilon. If we say approximate that
near zero then the first term in the tailor series of arch sawing
would be a2- a1 over two 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. 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 sine 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, like 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 would happen 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,
you may wonder when you're ever gonna 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,
anytime 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. 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 couple
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.