cc >> Our speaker tonight
is Clint Sprott. And Dr. Sprott has
a bachelor's degree from MIT, which he obtained in 1964.
He got a Ph.D. in physics from the University of Wisconsin
in 1969. And his professional interests
are, and I have to read this to get it right.
In experimental plasma physics and non-linear dynamics.
But he's also well-known for 25 years producing educational
programs under the title of Wonders of Science, which has
been on cable television, and available in lectures and
demonstrations at the University of Wisconsin and elsewhere.
So the title for tonight, is the "New Science of Chaos."
And I can make the snide remark that anybody who thought this
was going to be a talk about American foreign policy
in the 21st century, will be surprised,
but probably not disappointed! ( laughter )
( applause ) >> Thank you.
I'm going to talk to you today about my research, which is
about chaos, in perhaps more ways than one,
but I'm actually not going to say much about the research
I'm doing currently, because the topic itself is of sufficient
interest, that I would like to give you a little bit
of a tutorial and explain what chaos is all about,
and why some of us are devoting our careers to it.
I'm going to use some software that I'm wrote many,
many years ago, called "Chaos Demonstrations"
to illustrate many of the points that I'm making today.
It will take just a moment to switch screen modes
from time to time, but I'll try to make it quick.
If you go out into the real world and measure something,
more often than not, what you will see is something
that looks like that. Now this could be lots
of different things. If you're an economist,
this might be fluctuations of some financial market.
If you're an ecologist, it might be fluctuations
of some plant or animal species out in the wild.
If you're a meteorologist, it might be fluctuations
of the rainfall, or snowfall, or temperature data.
If you're into medicine, it might be the signal you get
from putting an electrode on someone's head
and looking at the fluctuations of their brain waves.
This could be lots of different things.
Until rather recently, just in the last few decades,
when scientists were confronted with something of that degree
of complexity, they tended to throw up their hands and say,
there's no hope of writing down some mathematical equation
whose solution is going to look like that.
The best we can do, is to sort of study it statistically.
Maybe learn some characteristics of it, but it is far
too complicated to model in any simple way.
Well, what I'm going to show you today, is that there are
many systems that are extremely simple that behave like that,
and there are many simple models,
some involve mathematics, some don't even go that complicated,
that produce behavior that looks very much like that.
Of course, not everything in the real world is not like that.
Many things in the real world are highly predictable.
An example is a pendulum. If I take this ball and let it
swing back and forth, I can predict with a high degree of
accuracy where it's going to be, at any time in the future
given the initial conditions. And I can also draw a graph
of it, and it simply goes back and forth, back and forth,
in a very regular, periodic manner.
And there are many things, of course, that are like that
in the real world. But not everything is.
And I'd like to show you another example on the computer here.
That's the example of a planet orbiting a star.
Now this is one of the first problems ever solved in the
modern sense in which we think of solving a problem in science.
It was solved almost 400 years ago by Johannes Kepler,
an astronomer, who was analyzing data from his mentor,
Tycho Brahe. And he discovered that
the planets orbit the sun in elliptical orbits.
And he was able to deduce many of the mathematical properties
of those ellipses. But the important thing from
our perspective is that this is a highly predictable
kind of behavior. And indeed the earth orbits
the sun in an elliptical orbit, it's very nearly circular, so
this is more like Hally's Comet, or something, but the important
thing is that we can predict where the earth will be
many years in advance. In fact, we're so good at
predicting things in astronomy, we can even predict when an
eclipse of the sun will occur, let's say, years in advance.
And we can even predict where on the earth to go, to stand,
to get the best view of that eclipse, many, many years ahead.
And that's truly remarkable that the equations of physics
known 400 years ago, or 300 years ago, anyway,
are sufficiently accurate and precise, and predictable,
that we can make those kinds of long-term predictions.
But my astronomy friends tell me that if you look
throughout the universe, most stars occur not as
single stars, like our sun, but as binary stars, or even
groups of three or more stars. You might wonder what happens
when a planet orbits, not a single star,
but a pair of nearby stars. Well, on the computer,
we can simulate that. So here are two stars,
they're actually at the center of these two circles.
Here's a planet that we started out in sort of
an arbitrary way, and you see it's going in a much more
complicated kind of orbit. And it looks almost random.
It looks like it's doing whatever it pleases.
Now if you think about it, this is truly remarkable,
because the underlying laws of physics are exactly
the same in these two problems. They both involve laws
handed down to us by Isaac Newton 350 years ago.
They involve the universal law of gravitation, any two objects
are attracted with the force that's inversely proportional
to the square of the distance between them, and they
accelerate in proportion to that force.
So with those simple mathematical equations, one
could write down the equation that is governing this system.
And it's basically the same equations in the two cases,
but yet the behavior is somehow qualitatively
very, very different. And this is an example,
the first example of many I'm going to show you
over the next hour of what we mean by chaotic motion.
So one characteristic of chaotic motion, as you might
gather is that it's complicated. Unlike the simple ellipse,
this is a much more complicated kind of motion.
It may also surprise you to know that the first problem
that I showed you, the so-called two-body problem,
a planet plus a star, was solved 400 years ago,
but the three-body problem, which I'm showing you here,
has never been solved. And probably never will be
solved in the sense in which we think of solving problems
in science. That is to say, there is no
mathematical equation you can write down that
allows you to predict where this planet will be at some
distant time in the future, in the same way that you can
in the two-body problem. And so it would be hopeless
to try to predict an eclipse in this case, more than a year
or two ahead. So it's a very complicated
motion, and it is unpredictable.
And that's one of the characteristics
of chaotic motion. Even though it's obeying
simple equations, simple laws of nature, we cannot predict,
at least for the long-term, what is going to happen.
We can predict the short-term, but if you try to go
too far into the future, your predictions fail miserably.
Now another characteristic of chaos, besides the fact that
it's complicated, and it doesn't repeat, obviously, is that it
has extreme sensitive dependence on initial conditions.
Now what does that mean. When I started this simulation,
I started the planet right here and I let it go,
and it did its thing. You might wonder what would
happen if I would repeat the calculation, but start
the planet in a little bit different condition, maybe
a velocity that's different by one part in a thousand.
Well, with the computer, we can actually do that.
And that's what this shows. Here, I'm actually doing
two calculations and just overlaying the result.
So we have two planets here, but the planets are not interacting.
They're just interacting with the two stars, which by the way
are not moving, so this is, in fact, a simplification
of the real three-body problem. And you see that for a little
while, the planets follow one another, but after a while,
they begin to separate. And if you watch it longer
and longer, you see that this separation grows ever larger.
After a little while, one planet is here and one is way over here
somewhere. So you might say, well, this
means that you have to know the initial condition
to a high degree of accuracy, maybe you need to know it
not one part in a thousand, but one part in a million.
I can tell you that if you repeated the calculation with
the initial condition different by one part in a million, by
this time, they would still be doing something different.
And it's because the error in the initial condition
grows exponentially. Now the thing you probably all
know about exponential growth, is that if you wait long enough,
it gets really big. It doesn't matter
what you're talking about, and that's the case here.
Do you have a question? >> You don't have an equation to
solve this, right? >> You do have an equation.
You can write down the equation for this, and it's no more
complicated than the equation for the two-body problem.
And of course, in a sense, the computer is solving
the equation, right? Otherwise, I couldn't
be making the graph. But the computer is solving it
in a special way. It's not solving it in the way
we usually think of solving problems in science.
It's not taking an equation, where you plug in a value for
the time on the right hand side of the equation,
and the left hand side then tells you where
the planet's going to be. What the computer is doing,
is inching itself along, in tiny little steps,
because you can always predict what's going to happen
in the immediate future. But what you can't do is make
a long-term prediction, without going through
all of the intermediate steps. It's those intermediate steps
that allow the error to accumulate and multiply,
and to eventually completely dominate
the behavior of the system. Now another characteristic
of a thing like this. Would you expect this to be
a good system to look for interplanetary life
or extra terrestrial life? The seasons would be pretty
extreme, wouldn't they, because at times you come
very, very close to the stars, and at other times,
you're much farther away. So that's a message
to the people who look for extra terrestrial life.
Don't bother looking around binary star systems.
That would be one message of this, if you want to know
what it's good for. That's one example.
Now, of course, this is a very old problem.
The problems in astronomy, they've been solved long ago,
the ones that can be solved, and the ones that can't be solved,
this is the best we can do. But there are many other
examples of things that have very predictable behavior.
I was about to show you this pendulum that swings
back and forth. And of course, if you think
about it, all of the methods we use to tell time,
basically rely on some sort of periodic oscillation,
even our very best clocks, atomic clocks,
rely on the oscillations of cesium atoms, or some other
kind of atom. So, it is some kind of regular
periodic oscillation that allows us to measure time at all.
And so it is in that sense that a periodic system, a regular
simply behaving system like that,
is highly predictable. Of course, another example
of that is this gadget right here.
Those of you who are musicians will recognize this as
an upside down pendulum. It's a metronome.
It beats a very regular pattern. It's very predictable.
If you listen to a few beats, you'll be able quite accurately,
to tell me when the tenth beat will occur.
Do you have a question? >> A little bit back to what you
said, that it would be possible to predict in the near term,
small, little, tiny predictions, but not in the long-term,
far out, and that's what the computer does.
Do they have any, or you have any theory, or otherwise,
an idea of why that is so? Is it just a fact,
without any understanding of it? >> There is some degree
of understanding. You're asking how do we know
when the behavior is going to be like this, and when it's
going to be like the other one. How do we know when the system
is going to behave chaotically? That's a question that doesn't
have a universal answer. We know some of the conditions
that are necessary for this kind of behavior, but even
when those conditions exist, you can't guarantee
just looking at the equations that they're going to have
this kind of behavior, without actually putting it
on the computer and trying it. So there is a sense in which
this is an experimental science. We're still grappling
with trying to understand the conditions under which systems
exhibit this kind of behavior, and the conditions under which
they don't. Some things are known about it,
but by no means everything. It's an active area of research.
Okay? Now with the pendulum,
of course, highly regular, highly predictable.
But I can modify this pendulum in one small way.
I have some little blue gadgets and I'm going to place them
underneath the pendulum and let it swing again.
Now if you look at the pendulum, especially if you could see it
from above, you will notice a very different kind of motion.
It's not simply swinging back and forth, it is wandering
around, not unlike what the planet does in the vicinity
of the two stars. And in fact, those blue things,
as you might have suspected, are magnets.
Two of them have their north poles up and two have
their north poles down. And inside the tennis ball
is another magnet. It's attracted to two of those
magnets and repelled by the other two.
And so in a sense, it is very similar to the problem here,
of the planet attracted to two stars.
And it sort of can't decide which star it wants to be near.
It's near one for a while and then it's near the other
for a while. And so that is one
of the characteristics. You have a sort of a tension
going on mathematically between two competing forces,
you might say. And that's a little bit
of an answer to your question. One of the sorts of things
you need to have for chaos. And if you just have two things
interacting, that is not sufficient, usually, to produce
this kind of chaotic behavior. Now you may conclude from this
that there's something mystical and unusual about
the magnetic force. It does have a certain mystique
about it. But, in fact, the equations
that describe the motion, are not particularly more difficult
when there is a magnetic force involved when there is only
a gravitational force. So it's not that
there is something mysterious about magnetism.
It's that it introduces an element into the mathematics
that allows the possibility of chaos.
And we think we understand what it is that does that
in the equations. You can understand it,
mathematically, in some sense. Well those are pendulums and
there are many other examples of pendulums of various sorts.
Here's another pendulum. This is a gadget you used to buy
at airport gift shops. I think I bought this one
for $8 about 20 years ago, I guess, now.
And normally, you buy it and you put it on your desk
and you stare at it and day dream.
But, it illustrates some important principles.
First of all, it looks like perpetual motion.
The thing's been going there ever since I started talking,
and it never seems to die. But of course, there is no such
thing as perpetual motion. There's a little magnet in
the bottom of that outer ring, and when it sweeps by the base,
there's an electromagnet there that senses that it's coming by,
and it gives it a little kick. So it's like someone's pushing
the swing. So that outer ring is going
back and forth in a regular periodic manner, approximately,
much like the pendulum before I put the magnets
near it. But if you look at that
inner ring, it's going around in a much more complicated
kind of fashion. In fact, it is exhibiting
chaotic motion. If I had two of these gadgets,
and I set them up side by side, much as I did here,
tried to start them identically, they would follow each other
for a couple of swings, and then they would get
out of step. Pretty soon, they would be doing
completely different things. So that's an example of chaos.
And, in fact, when I bought this gadget, there was some
extra magnets on the inner ring and the outer ring, so when
they came by each other, they got a little extra kick.
And I reasoned that those extra magnets weren't really essential
for the operation of it, and so I ripped them off.
And in a great triumph of theoretical physics,
the thing worked more or less the same.
( light laughter ) I suspected that it would work
because of, in this case, we know the reason,
the inner ring is chaotic and the outer ring is periodic.
And it's because the inner ring is not oscillating
to a very large amplitude. It's just going back and forth,
maybe 20 degrees or so. Sorry, the outer ring.
The inner ring, however, occasionally goes over the top.
And that's the critical element, that going over the top.
That brings into the mathematics something we call
a non-linearity. Now what does that mean?
It's a technical term, but it's one that comes up repeatedly in
those whole business of chaos. First of all, let's talk
about the linear case. The outer ring, when you
displace it away from its equilibrium down at the bottom,
there's a force pulling it back. That force is proportional
to the displacement. That's what we mean by linear.
If you displace it that way, the force brings it back,
but the larger the displacement, the larger the force,
and they are linearly related. So that's what linearity means.
The inner ring, though, when you start to go up
at a high elevation like this, the force doesn't
keep going up, up, up, up. In fact, when you get exactly
inverted, the force has gone all the way to zero again.
It's like a pendulum balanced on its point.
There's no force at all there. In fact, it is that
non-linearity that is responsible for the chaos.
So you've just learned something, in answer
to your question, in fact. A necessary condition
for a system to exhibit chaos is that there be some kind of
non-linearity in the mathematical equation.
So if you write down an equation and it has no non-linearity,
without any thought, you can say that can't do this.
Now that's not to say that every non-linear system
exhibits chaos. In fact, the two-body problem
is non-linear, because remember the force
was inversely proportional to the square of the distance.
That's Newton's law of gravity. And that's a non-linearity,
one over the square of something is a non-linear force.
So mathematicians say, non-linearity is a necessary,
but not a sufficient condition to chaos to exist.
So you've learned a bunch of things about chaos already.
You've learned that it's complicated.
You've learned that it doesn't repeat.
You've learned that it is sensitively dependent
on initial conditions. And you've learned that
it requires a non-linearity. So you're getting to be pretty
expert on this by this time. >> By chaos, do you mean
unpredictable? >> Well, I just defined chaos,
in essence. That is one aspect of it.
In the long-term, it is unpredictable.
That's right. You can predict the near future,
because you know the equations, you can write them down
and figure what's going to happen next.
But in the long-term, it's unpredictable.
And it's unpredictable because of this sensitive dependence
on initial conditions. Now, this was first discovered--
Yeah, back in the back. >> The near-term predictions,
which you say are accurate, are not really accurate,
they're only approximations. >> Well, that's right.
They're typically never better than the error
in the initial condition. As you go forward in time,
that error grows exponentially. So for a while, it's not
very big, but then eventually, it's enormous.
But it's never completely accurate.
Never completely accurate. Well, it may happen to be.
In fact, I can give you an example.
If I unplug the battery that's keeping this going,
I know the long-term solution to this.
( light laughter ) It's going to be sitting there,
not moving. So it's not quite true
that we can never predict. There are examples of what
we call transient chaos, where something goes through
a chaotic behavior, but it eventually
settles down to something that is quite simple
and quite predictable. And so, there are many examples
of these things. Now here's another pendulum.
This is actually a double pendulum.
It's a pendulum supported by another pendulum.
You're going to have trouble seeing this, aren't you?
I can hold it up a little bit. So it's two pendulums.
And you can see a complicated kind of behavior,
if I start is swinging. The upper pendulum
behaves in a fairly simple way. The lower pendulum
is much more complicated. But this is an example of
what you were just asking about. I know, eventually,
what it's going to do. It's going to be hanging
just like that, no matter how I start it.
But yet, in order to get there, is goes through a chaotic
kind of motion. Now of course, it comes to rest
because of friction. And if I could build
this thing without friction, then you might expect it
to go on chaotically forever. Well, it's very hard to do that,
but what I can do is add energy to the system
at the same rate the friction is removing it.
And that's in fact, what's happening over here.
That mechanism that's keeping the thing going is adding energy
at the same rate friction is taking it out.
So let's see if I can do this while holding it.
I'm going to just swing the upper pendulum back and forth
in a regular manner. And you see the upper pendulum
is now swinging in a regular way and the lower pendulum
is now swinging in a chaotic manner.
So it's really the same example I showed over here.
The upper pendulum is not swinging
to a very large amplitude. It's behaving
in a linear fashion, and can't exhibit chaos.
The lower pendulum can go all the way over the top,
and so it has a strong non-linearity that causes it
to behave chaotically. So those are some examples
of chaos in pendulums. Here's another curious
kind of pendulum. Here I have an upside down
U-shaped gadget. It has two arms on it.
One arm is free to swing back and forth,
the other arm is rigid, and it also pivots in this way.
That's means there are basically two kinds of motion.
One kind of motion has this relatively fixed and
this swinging back and forth, like so.
There's another kind of motion where this stays more or less
up and down and the whole thing does that.
But the two motions are obviously coupled together
in a fashion. And so if I start it up
like this, give it a lot of initial energy,
you see those motions cause it to undergo a very complicated
kind of behavior. Now this is a case where I could
build two of these devices, try to make them as identical
as our shop is able to manufacture them,
put them side by side, start them exactly the same,
as close as I'm able to do. And you can well imagine that
they would stay together for a little while, but eventually,
they would be doing totally different things.
However, if you wait long enough, because of the friction,
you know what the final state's going to be.
So this is another example of what we call transient chaos
and if there were no friction, it would go on forever.
It wouldn't be a transient, or if we somehow put energy
back into the system, that would keep it going forever.
So these are examples of chaos in mechanical systems.
Now this idea of chaos is really not as unfamiliar
to you as you might think. One of the very first things
we teach our physics students is the way things fall
under the influence of gravity. And sort of in the first week
of the first course in physics, we teach people that when
you drop a ball, it falls, and it falls straight down.
And we teach them how to calculate from the equations
of Newton, how long it takes to hit the ground, how fast
it's going when it hits, where it is at any instant
after I release it. And I can even do it
if I throw it up in the air with some initial condition.
You can solve all that, if you've had, even,
let's say two weeks of an elementary physics course.
That's a very simple problem, one of the things
we teach students. However, if instead of doing it
with that rubber ball, I use this little index card
and drop it. That time it dropped
pretty straight. Let's try it that way.
You can see a very different kind of behavior.
Now you can well imagine that if I had a hundred of these
index cards, and I started them all as close as I could
to the same condition and drop them,
there would be cards all over the floor here.
And that's because the motion, as it falls is chaotic,
it flutters in a way that is very sensitive
to the exact starting condition. Now that's curious
when you think about it. These two objects are obeying
exactly the same laws of physics.
They're both being attracted to the earth by their weight,
their gravity. And they both have some
air resistance, but yet the motion is somehow qualitatively
completely different in the two cases.
One is regular motion. One is chaotic motion.
It may surprise you to know, it's only been in the last
ten years or so, that someone has actually written down
the mathematical equation for that object falling
and demonstrating mathematically that indeed it should be chaotic
and finding the conditions under which it becomes chaotic
or becomes regular. There is a critical value of
the size, as you might imagine, and the shape of it that allows
it to behave chaotically. Now another thing
we teach students, we teach them about rockets.
You know, physics is rocket science.
And in the textbook, there are all these nice pictures
of what happens when the exhaust comes out the back end
of the rocket. The rocket goes that way
and there are equations that tell you how to predict
how fast the going to go, how far it's going to go.
Nice equations. But when you actually try
to do it with a real rocket, I don't have a rocket,
but I do have a balloon. I'm going to try this one.
So of course, this is a rocket. There's some gas in there,
and when the gas comes out that way, the rocket's
supposed to that way, and the book shows how that works.
( laughter ) But nowhere in the textbook
does it tell you about all of that.
At least not in the textbooks that we use to teach
our students at this university. And that's because
this is a chaotic motion. When it starts to leave,
it doesn't go in an absolutely straight line.
A slight departure from straight grows.
And in fact, that error in the starting position
grows exponentially. Now you can imagine if I
repeated this with 100 balloons, tried to start them all
exactly the same, there would be balloons all over the room
up here, after a while. I don't need to do the
experiment for you to realize that's the case.
That's an example of chaos. Chaos is sort of all around us,
and we ignore it. We somehow say,
oh, let's not think about that. We like to solve problems
we can write the equations down and predict
what's going to happen. And in fact,
we teach our students that. And so, I've told you something
a little bit remarkable. I've told you that
we as educators, as physicists, in particular, have sort of
swindled our students, and science in general
has done this. We have led students to believe,
and maybe you share this belief, as I did until very recently,
that the essence of science is predictability.
And if you can't predict what's going to happen, you really
aren't doing good science. What I'm telling you is there
are situations, and they're not at all uncommon,
where your best effort to make a prediction
is doomed to failure. You will never predict.
Another thing we think is basic to science,
to the scientific method is the repeatability
of an experiment. You're supposed to be able to do
an experiment several times and get the same answer each time.
And if you can't do that, then you're not doing science.
That's sort of inherent in what we think science
is all about. But yet I'm telling you
there are lots of examples of simple things in science,
even in physics, which in some sense
is the simplest of the sciences that deals with things
that don't even have a brain. And yet, you try to do the
experiment twice and you get different answers,
or ten times and you get ten different answers.
So it's a remarkable adjustment in our way of thinking.
A sort of a paradigm shift that we have to begin to see
science in this broader context. That there are things
we can't predict, and that not all of our experiments
are, in fact, repeatable. Now let's talk about
this pendulum a little bit. Yeah, you have another question?
When you say, starting out those cards or something,
exactly the same, I stop at the word "exactly."
Did I say exactly? >> Does that mean "exactly"
as far as we can make it the same, does it mean identical?
Yes, I mis-spoke. When I said exactly,
I meant exactly to the degree that we are able to do it.
Even if we could start them exactly the same,
if the cards weren't exactly the same, they would
have different behavior. Lots of things have to be exact,
besides the initial condition, the parameters that govern
the system itself, have to be exactly identical,
or you will get this behavior. It's a very good question,
and thanks for correcting me. Okay now, you may have
drawn the conclusion that chaos is complete disorder
and randomness. And I want to disabuse you
of that idea, because all of these examples I've shown you,
in fact, can be described in terms of very simple equations
that you can write down. And the equations don't have
any randomness in them. And you know at some level
this cannot be a random process. It is being governed
by what we call determinism, that the future is uniquely
determined by the present. It's just that it has this
peculiar characteristic that the initial conditions grow.
And so, it would be nice if we could examine a thing like that,
and convince ourselves that this indeed is behaving
chaotically and not randomly.
So I'm drawing a distinction. Something I haven't done before
between a chaotic process and a random process.
They're different things. The chaotic process,
for one thing, is predictable over the short-term,
whereas a random process is not necessarily.
Flipping a coin, you don't know the next step, whether
your going to go right or left, if you're deciding
by the flip of a coin. But if you're deciding
by an equation, you know the next step,
which way you're going to go. And so randomness and chaos
are two different things, but they have a lot in common.
So if you see something like that in nature,
how do you know that it comes from some simple equation
that happens to be chaotic, versus some random process,
the Gods flipping coins and deciding
where you're going to go next. Well, there are ways to do that.
Let's go back to the simple pendulum.
I tried to convince you that the pendulum, if it swings
through a small amplitude is periodic,
but if it swings through a large amplitude,
it is potentially chaotic. So in such a system, if you
imagine this pendulum swinging back and forth,
being pushed regularly to overcome the friction,
but the push is very regular and repeatable.
That's important. If you're pushing it randomly,
you would expect it to behave randomly.
But the idea is in this outer ring,
we're pushing the inner ring in a regular periodic manner.
Okay, now imagine that we look at the lower pendulum
over there, or the inner ring here.
And we ask, well where is that inner pendulum
when the outer pendulum reaches a certain position.
Like when it gets there, we could ask
where the inner pendulum is. And we could do that,
either experimentally, or we could simulate it
on the computer. And what I'm going to do is
simulate that on the computer. A pendulum being driven
periodically by some external force.
Can you actually turn the lights down from back there
a little bit? This is hard to see otherwise.
So here we have a pendulum. Now I can't see
where I put my pointer. Anyway, here's a pendulum,
simulated on the computer. It's a very simple set
of equations that most any freshman physics student
could write down, except to solve them you have to
put them on the computer. So here's a pendulum
being driven back and forth. It's got some friction.
The friction is being overcome by this source of energy.
So here we have a pendulum that has some friction,
but it also has a periodic push back and forth, so it's behaving
in a regular periodic manner. Now as with many of these
demonstrations, I can push just a little bit harder.
And if I do that, the pendulum will in fact,
go to a large amplitude, and even occasionally
go over the top. Now you have an example of
what we mean by chaotic motion. You notice that the equations
that govern the system have not changed.
All we've changed is some parameter that is represented
by how hard you're pushing. If you push hard enough,
it behaves this way. If you don't push hard enough,
it is regular and repeatable. And so, this is another
characteristic of chaos. Chaos typically has associated
with it a parameter, a knob, if you will,
that you can adjust, and for some values of that knob,
you get regular behavior, periodic in this case.
For other values of that knob, you get chaotic behavior.
And so that's, I guess the fifth characteristic of chaos
that I've told you about today. Well, of course, just staring
at a thing like that, would never convince
a reputable mathematician that this is chaotic motion.
You have to do something a little more than this.
And one thing you could do is make a graph of something.
Well, you could graph the angle that the pendulum makes
versus time. If it's periodic, it would
just swing back and forth, the angle would go up and down,
up and down, up and down. Let's see what this one
is doing. I've sped the calculation up
a factor of a hundred, or so, so you get lots and lots
of these cycles to look at. And you see they never
obviously repeat. It goes back and forth,
back and forth, but not in any evident way.
You could also look at the speed with which the pendulum
is moving. If you do that, you get
another complicated plot that doesn't obviously repeat.
Now it could be that it repeats, but it just takes a very long
time, and you've forgotten what it did before,
once you see it again. So a better way would be to
make a plot, on which the speed is on one axis, and the
position is on the other axis. And such a plot, which the
computer has been calculating, look the something like this.
Now if this were a regular motion.
This would be an ellipse. It would simply be going around
and around in this funny space of the speed of the pendulum
and the angle of the pendulum. It would be always repeating
what it did in the previous cycle.
But here you see a much more complicated kind of motion.
It looks like it can go pretty much anywhere
it darn well pleases. And if you wait long enough,
it will be anywhere. At least within these bounds.
And you may wonder why a chaotic system has such a rigid upper
bound here and here. And that's one of the clues that
this is not a random process. This is a chaotic process.
It's obeying some rules. And you can see one of the rules
is that you can never go lower than this
or higher than that. And so this is one way
of diagnosing a system to see that you're seeing evidence
of chaos. Now remember what it is.
This is a pendulum going over the top.
And I'm plotting its speed and its angle
at every instance of time. Suppose, instead of plotting
at every instance of time, I only plot at certain
instances of time, like when the thing that's driving it.
If this were the lower pendulum here of the double pendulum,
whenever the upper pendulum gets to a certain angle,
I would then plot the speed and the position
of the lower pendulum. Clearly those dots would be
somewhere on these curves, but would that plot
fill in the whole space? Well, let's look and see,
because the computer, in fact, has been doing this.
You can see, indeed, there are lots of dots here, and they
do fall within those bounds that I showed you before,
but now you see a more intricate structure.
You see an object that goes by the very wonderful
mathematical name of a strange attractor.
It's a wonderful name. And it's an attractor
in the sense that you can start this pendulum anywhere you want.
You can start it up here. High velocity at a certain
angle, and you watch it for a little while,
and it goes over to here, and then it moves around
always staying on this object. And that's the sense in which
it attracts different initial conditions.
Now another thing you notice about this, is that it's
sort of a complicated thing. In fact, this is an example,
this strange attractor is an example of something
mathematicians call a fractal. And it's hard
to talk about chaos without talking about fractals.
Fractals are sort of to chaos what geometry is to Algebra,
if you will. A fractal is
the geometric manifestation of the chaotic dynamics.
And so if you plot something in the right kind of way that is
chaotic, you will typically see such a fractal object.
Now there's a whole lecture I give on fractals.
It's a fascinating subject in its own right, but I'll just
give you the one-minute version. A fractal is a mathematical
object, like a plane or a sphere, that has
a peculiar characteristic of being self-similar.
Remember back in high school in geometry, you learned
about similar triangles? They were the same shape.
They had the same angles, but they were
just different sizes. Well, a fractal is an object
that is self-similar. That means it contains copies
of itself and not just on two scales, but on all scales.
Which is to say, that if you zoom in on a piece of it here.
What looks like a little band here, you discover
are really a bunch of lines, or a bunch of bands
with gaps in between, and you can see some of the bigger
gaps here, in fact. If you zoom in on this
and then blow it up so it's as big as this,
you see this little gap then becomes big again.
And as you zoom in deeper and deeper into a fractal object,
you continue to see structure. And the structure resembles
the structure you saw before you zoomed in.
And so, you've learned something not about chaos,
but about a fractal. Now a fractal also has
another interesting property. It has a dimension
that is not an integer. Now what does that mean?
Well, you know that there are various kinds of objects that
you learn about in geometry. A point, or even a set of points
has zero dimensions. If you have a line,
even an infinitely long line, it has a dimension of one.
That is to say, you can specify a length, you can specify
where you are along the line with a single number.
A surface, like a piece of paper to a good approximation,
is a two-dimensional object, or a ball is a three-dimensional
object. You're used to the idea, and
you've been trained this way, as I was, that all objects
in nature have dimensions zero, one, two, three.
And in fancy mathematics, you might have hyper-volumes
and things, and four dimensions. But here is an object
that doesn't neatly fall into either of those categories.
This is somehow more than a very long line
that has been wrapped around lots and lots times,
but it's less than a surface that has some holes
cut out of it. It has a dimension, in fact,
that is greater than one, but smaller than two.
And this particular object, you can define the dimension of,
has a dimension of about one and a half.
There's a way to calculate these things.
And there's a whole mathematics that's been built up around
fractal objects, because it turns out
if this looks like something you might have seen in nature,
I'll come back to that in a moment,
there's no accident to that. Because nature is full
of objects that have fractal characteristics.
They don't satisfy the mathematical definition
exactly, but they do have this characteristic
that as you zoom in, you see smaller and smaller
copies of it. A tree is an example.
If you look at the limb of a tree, it looks a lot like
the trunk. And the limb has limbs,
and those limbs have limbs, which have limbs,
which have limbs. You get the idea.
Self-similarity in trees. Or a coastline.
If you zoom in on a coastline seen from a satellite,
let's say, in deep outer space, as you get closer and closer,
you see more and more structure. It has a kind of a statistical
self-similarity. Anyway, those are fractals.
And I won't digress any more, except to say that looking for
fractals, in a dynamical system, as with the pendulum like this,
is an excellent way of determining that you are
looking at a chaotic system, rather than a random system.
Because a random system typically will not produce
such a fractal pattern. Now remember what this was.
These are dots that represent the position of the chaotic
pendulum at a particular point in the oscillation
of the non-chaotics thing that's driving it.
The periodic system that's driving it.
We could, of course, have made a plot at various positions
of the driving system. And one thing you know
is that if you're driving it in an irregular periodic manner,
is that a succession of such movies, or
a succession of such pictures, will eventually come back
to where it started. And so, you can make it
kind of a movie. And one thing you know
about the movie without doing it is that it's going to repeat
endlessly. Well, the computer's
been collecting data for this. And here's what it's doing.
This is like 30 frames, I think. It's kind of slowed down.
I'll see if I can make it go faster here.
The computer is showing about 30 frames of this movie.
And you can see, it keeps coming back to where it was.
Now does this look like something any of you have seen
in nature somewhere? Somebody said waves.
It looks like waves. Anything else?
Whirlpool, yep. Snail.
Those are all good. You know what it reminds me of?
Have you seen those taffy machines that stretch
and fold the taffy? Oops, now I've done it,
but that's okay. I'm basically done
with this anyway. Okay, if I can have
the lights back up. It reminds me of those taffy
machines that stretch and fold. In fact, this is an excellent
analogy to what's going on in a chaotic system.
And I don't have some taffy. But I do have some silly putty
right here, that I bought down at the toy store.
If I work this like taffy, I stretch it and I fold it.
The taffy machine does that over and over, and over again.
I'm not quite sure what they're doing, but anyway
that's how you make taffy. Now, you notice I've made
a kind of a horse shoe thing when I stretch it and fold it.
So imagine that I do that, and now I do it again.
If I do it a second time, you see the horse shoe
has now become two smaller horse shoes.
A little hard to hold it in my hand,
and we have a bigger horse shoe. So this is how all of that
self-similarity comes about. Every time I stretch and fold,
I take the structure that occurs on the largest scale
and I drive it down to one smaller level.
And so, if I do this a thousand times,
the structure of the first, has gone down a thousand levels.
If each one gets smaller by a factor of two, it's a
power of two to the thousand. It's down to a really,
really tiny level. So that's where the fractal
comes from. From the stretching and folding
of the so-called phase space. Now I can use this
to also illustrate how this comes to be chaotic.
Remember these represent all the possible positions
and velocities of a pendulum swinging back and forth.
I could put two marks on here. You won't be able to see them,
but imagine I put two nearby dots and those represent
the initial conditions. And let's say I start
on the attractor. It doesn't matter where I start,
but for simplicity, let's start there.
Now I stretch it. Well, they're now twice as far
apart as they were. When I fold it, they're still
twice as far apart as they were. I stretch it again, they're now
four times as far apart. Eight times, 16 times, 32 times.
You can see where the error in that starting condition
is growing exponentially. It's a very simple geometric way
to show where the error in initial condition comes from.
Now I just have a couple of other demonstrations.
Here's one of my favorites-- Well, let me show you this one.
I like to do things with balls. I showed you a tennis ball.
I showed you a super ball. Here I have a ping pong ball.
I'm going to put this ping pong ball in a little petri dish.
And the petri dish is mounted on the cone of a loud speaker.
You don't actually need to see this, just listen.
That's all you need to do. What I'm going to do is make
the speaker bounce up and down. I've got it connected here
to an audio oscillator that goes up and down
eight times a second. Eight hertz.
It's a little below what you can hear with your ear,
but you will be able to hear the ball bounce.
Now you notice there's a knob. And the knob I'm adjusting
is the amplitude or the volume of that eight hertz sound
that's going into the speaker. Now what I want you to notice
is that under some adjustments of that knob,
it's a little hard to find one right now.
For some values of that knob, it's like the metronome.
Maybe the table's a little unsteady,
that may be the problem. Let me put
the microphone closer. ( rhythmic tapping sound )
Very regular and repeatable, but a small change in this knob,
makes a very different kind of behavior.
( random tapping sound ) So that's another example
of a chaotic system having a knob that you can turn,
and for some values of the knob, it's very regular.
For other values, it's completely chaotic.
And the transition can be very abrupt.
Now I want to show you another example of the same thing.
And this is one of my favorite demonstrations, because you
can all go home and repeat it. You don't need
any fancy equipment. All you need is a faucet in
your kitchen or your bathroom, that you can open a little bit.
So here I have a beaker with water in it.
And here's a faucet. I have a little pie tin here.
And when it drips on this, you're going to hear it,
when I put the microphone down next to it.
I'm going to open the faucet and let it drip,
first kind of slowly.
And then I'm going to put the microphone down there
and let you hear the drips. ( soft, rhythmic dripping sound )
Can you hear that? Couldn't quite hear that?
Let's see, what can I do? Let me drip it
at a larger distance. I'll put the plate
all the way on the floor. There we go.
( rhythmic dripping sound ) Okay very regular,
very repeatable, right? You probably all have a faucet
at home that does that, don't you?
( laughter ) It probably keeps you awake
at night. Well, if that happens,
here's an experiment you can do. Go and adjust the drip rate
of the faucet. Open it a little bit
to make it drip faster and see what happens.
Let's see if we can do that. ( random dripping sound )
So here we have another system that has a knob, the drip rate.
And for certain adjustments of that knob, it's very regular,
very predictable. For other adjustments
of the knob, it seems to have no pattern.
It seems to be going randomly. But, in fact,
it's an example of chaos. If you do this experiment at
home, here's something to try. Explore the transition where
it goes from regular, periodic, to chaotic, and see if you can
hear the following. What you should hear,
as you turn the drip rate up. First of all, it will drip
faster, obviously. Instead of drip, drip, drip,
it will go drip, drip, drip, drip.
That's not very interesting. But you will reach a certain
point, and I won't try to do it here.
It's a little tedious. But you'll reach a point
where every other drip repeats. It will sound something like
drip, drip-drip, drip-drip, drip-drip, drip-drip.
Have you ever heard a faucet do that?
I bet you have, if you've paid attention.
At certain critical value of the drip rate, it will repeat
every other drip. Now here's something
even more challenging. Turn it up a little bit more.
What will happen, is it should repeat every fourth drip.
It sounds something like drip, drip-drip-drip,
drip, drip-drip-drip,
something like that. This is what's called
a period doubling bifurcation. In the language of chaos.
But you can see what's happening.
Instead of repeating every drip, it repeats every other drip and
then every fourth, eighth, 16th. It turns out that the amount
by which you have to turn the knob is ever smaller
when you go to higher and higher periods.
And you eventually reach a point where it's essentially
infinitely many drips before it repeats.
And that's what we mean by chaos.
You've lost the periodicity. And so the period doubling
comes from the fact that first it's every drip,
period one. Then it's every other drip,
it repeats. Period two.
Then every fourth drip it repeats.
Period four. Period eight, period 16,
period 32, finally chaos. That's a very nice experiment
of chaos that you can do right in your home.
Now let's see, I have this one last example.
And that's chaos in electrical circuits.
Many years ago, I actually got interested in physics
by being a ham radio operator. And when I was a teenager,
I used to build all kinds of electronics.
And so when I got to college and became a professor,
they assigned me to teach the electronics course.
I thought I was really sharp at this.
I had been doing this since I was literally a baby almost.
And I even wrote a book, "Modern Electronics."
And I never knew that there were chaotic electric circuits
until about 20 years ago now. And they don't have to be
very complicated. Just like all of these
other demonstrations, if you set something up in the right way,
you can make something exhibit chaos without it
being very complicated at all. And actually, I have a chaotic
electrical circuit here. I don't want to get into
the details of what's in it, but suffice its to say,
it's a very, very simple electronic circuit.
In fact, there are many circuits that will do this,
that are now known. But this is one I'm particularly
proud of, because I invented it. And you can go out and buy them
for $600, they sell them for. The money doesn't go to me.
I get nothing. But it's an oscillator.
You've all the experience of having a microphone
too close to the speaker and you get feedback?
Audio feedback we call it, right?
And this is one way to make a system chaotic, in fact.
Normally, when a system undergoes feedback,
you hear a screeching sound. A periodic oscillation,
very loud, very annoying, you cover your ears,
but it's typically not chaotic. And that's because an amplifier
typically is made to be a linear device.
And at worst, it just eventually saturates
and doesn't go any louder. But it is basically behaving
in a linear fashion, and that's why there is no chaos.
Now there are many linear electrical circuit components.
Those of you who are into this, a diode that passes
electric current one way, but not the other way.
It's a one way valve, you might say.
That's a non-linear element and that's all you need.
A diode in the right kind of circuit can exhibit chaos.
I have a little circuit here, and I have it connected up
to an audio amplifier, so you can actually hear it.
So here is the tone that it's making.
I'll hold it near the microphone so you can all hear that.
( monotone ) So you can imagine that this
was a tone being made with a microphone too close
to a speaker and you're getting feedback to a particular
frequency. But there's a knob.
And the knob in this case is basically the volume control.
As I turn up the volume control, you will hear
something interesting. ( tone changes to lower tone )
Now, what was that? First you heard that tone
and then you heard that tone. Now those of you who are
sort of musical may recognize that the second tone
is just one octave below the first tone.
They're both kind of low, so it's hard to hear that.
That's the period doubling. We have a knob, we turn it up.
And instead of every cycle repeating, every other cycle
repeats, so it has two littles and a big.
Big cycle, little cycle, big cycle, little cycle.
Period two. And then you have
every fourth repeating. Let's see if we can hear that.
( tone goes lower, and then sounds like multi-tone static )
It's a little hard to hear because chaos onsets.
So that's the kind of feedback you would get in an audio
circuit that had the right kind of non-linearity.
You could produce chaos that way.
And so, I hope these examples its have convinced you
that chaos is all around you. That it violates many
of the principles that you've come to hold near and dear
about how science is supposed to operate.
That it gives us a new way of looking at the world when
we see complexity in the world, something behaving
in a complex way, we now have the hope that there
may be some simple explanation. That we don't have to appeal
to someone flipping coins or some random process.
That it may be that very simple models
can produce the complexity we see in the world around us.
Of course, there's a lot of interest.
People who want to predict the stock market and understand it,
would love to discover that there's some simple model
equations you could write down that would at least do
sort term prediction. You don't have to be very good
at it, just a little bit better than everyone else, and
it makes a lot of money for you. Or an ecologist who thinks that
they see some animal species about to die, and they say, oh,
this is terrible, there must be something we're doing.
It might not be anything we're doing.
It might just be inherent in the dynamics of the system
that things are going to undergo these fluctuations
no matter what. Or the weather, we attribute a
lot of things to global warming these days.
You see fluctuations of the snowfall this winter
and you say, oh, it must be global warming.
But maybe not, maybe there's an underlying chaotic dynamic
that if we could only understand it and model it
with some simple system of equations, we might learn
quite a bit about it, without having to appeal
to all of the causes and mechanisms
that people imagine doing. Now let me just close
with a commercial. If any of this stuff interests
you, every Tuesday noon over in Chamberlin Hall,
we have a seminar, open for anyone, called
"Chaos in Complex Systems." And we basically have people
come and talk, typically from around the campus, who are
working on systems of this sort, not necessarily chaotic,
but systems that exhibit complex temporal dynamics.
You're all welcome to come and attend.
There's only three or four more, maybe five or six more,
to the end of the semester. But if you're interested
in that, come and pick up one of these afterwards.
We would more than welcome your attendance.
So thank you very much. ( applause )
Can I take a question? >> Yes, I'm just wondering
what kind of mathematics would describe this system.
You can't use linear math. Is that abstract mathematics
they use to describe this? >> Funny, you should ask.
Let's see, I actually have an equation on here
that I was about to show. I can show you an example
of an extremely simple equation that exhibits chaos.
And it may look complicated at the moment,
but let me say what it is. Imagine you have some money
in the bank and it's earning interest.
And so, the bank is going to calculate your balance next year
by taking your current balance and multiplying by some factor.
Let's say you're earning ten percent interest.
So, next year, you have 1.1 times what you have
this year. The year after, 1.1 times that,
and so forth. Mathematically,
that's called iteration. And mathematically,
it's like feedback, because you're taking the new value
and using it as the input to the same equation
to get the next value, over and over and over.
It's a little like that silly putty, isn't it?
You're wrapping it around and around.
Or it's like the audio feedback. Something's going round
and round, but in the process, it keeps amplifying.
Okay, that's a linear system. Because what you have next year
is proportional to what you have this year.
Linearly related and no chaos. You can earn interest forever.
Your money is exponentially growing so that's nice,
but very predictable even over the long-term.
Now, something you all know is that something cannot grow
exponentially forever. Eventually, something's
going to stop the growth. The bank's going to run out
of money at some point. In an ecological system,
you're going to run out of food. Maybe we're going to do this
as a society at some point, and our population
is going to have to stop growing exponentially.
It actually has already, in fact.
The thing that stops the growth is typically a non-linearity.
Now the simplest non-linearity you could put in mathematically
is to say that, let's say the bank is going to do this
to keep from having you develop an infinite amount of money.
So the money you have next year is this constant 1.1, let's say,
times this year's money. But now we multiply it
by [1-x(n)]. So when X is a small number
compared to one in this case, it doesn't do very much.
It's a linear growth. But as soon as X starts
to approach one, this turns the growth rate down.
So mathematically, this is an equation that does something
very realistic. Anything that grows
exponentially, it eventually stops it, puts a clamp on it.
And this system is capable of exhibiting chaos.
That's all you need. You have to go up to a value
of A, that's about 3-1/2, between 3-1/2 and 4.
This equation exhibits chaos. It has period doubling.
It has sensitive dependence on initial conditions.
It has a non-linearity. Here's X times X, X squared.
Everything that I've been talking about is contained
in this simple equation. So yes, there are extremely
simple equations that exhibit this behavior.
That was a great question. Any others?
So does the chaotic system include--
it's inclusive of linearity and regularity.
It's inclusive of, but a little greater than that.
Right. Typically a chaotic system
has some parameter. In this case the A is the
parameter, the interest rate. If you turn that, you will
typically have values of it where it's chaotic
and values where it's not. So indeed a chaotic system
will almost always have conditions under which
it behaves non-chaotically. And somewhat surprisingly,
even amidst the chaos, if you have a region where
there's lots of chaos, you can turn that parameter
a little bit, often, and find a window of periodic behavior,
for a small range of parameters. In fact, for this equation,
there's an infinitely many such windows, even in a small
region of the parameter A. So it's fascinating
mathematically. And this is mathematics that any
high school student could do. This is a parabolic equation,
a quadratic equation. It's X depends on X
and X squared. So any high school student
should be able to do this. It's like a course in algebra.
You can program this on the computer and see
all of these things that I've been talking about
right here. Any other questions?
If you drop the index cards many times, and you recorded
the distances of the index cards from your feet, how far
they have fallen, could you get gaussian distribution, or would
you get a fractal distribution? >> That's a good question.
If you drop the index cards and then look at where they fall
and plot the distribution, you'd expect it to be peaked
right under your feet and then to die off.
I think in that case, you would see something like a gaussian.
I don't think you would see a fractal.
So it's not the case that everything you plot
is going to give you a fractal. Some things do,
but many things don't. >> So you might still get
a gaussian distribution by pooling together a lot
of observations, even if they are chaotic?
That's right. Chaotic systems can mimic
a random process in many, many ways,
and one way is that it can make gaussian distributions.
Thank you very much. ( applause )