cc
is Clint Sprott. >> Our speaker tonight
a bachelor's degree from MIT, which he obtained in 1964.
He got a Ph.D. in physics from
in 1969. the University of Wisconsin
are, and I have to read this And his professional interests
In experimental plasma physics to get it right.
But he's also well-known for and non-linear dynamics.
programs under the title of Wonders of Science, which hasl
been on cable television, and available in lectures and
demonstrations at the University
So the title for tonight, is of Wisconsin and elsewhere.
And I can make the snide remark that anybody who thought this
was going to be a talk
in the 21st century, about American foreign policy
but probably not disappointed! will be surprised,
( applause ) ( laughter )
I'm going to talk to you today >> Thank you.
about chaos, in perhaps aboumore ways than one,h is
but I'm actually not going
I'm doing currently, because the to say much about the research
interest, that I would like topto give you a little bitnt
of a tutorial and explain what chaos is all about,
and why some of us
I'm going to use some software are devoting our careers to it.
many years ago, called that I'm wrote many,
to illustrate many of the points "Chaos Demonstrations"
It will take just a moment that I'm making today.
from time to time, but I'll try toto make it quick.des
If you go out into the real world and measure something,
more often than not,
that looks like that. whatNow this could be lotshing
of different things.
this might be fluctuations If you're an economist,
If you're an ecologist, oit might be fluctuations
of some plant or animal species out in the wild.
If you're a meteorologist,
of the rainfall, or snowfall, itor temperature data.ns
If you're into medicine,
from putting an electrode it migon someone's headyou get
and looking at the fluctuations of their brain waves.
This could be lots
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
The best we can do, is to sort of study it statistically.
Maybe learn some characteristics
too complicated to model of it, but it is far
Well, what I'm going to show in any simple way.
many systems that are extremely simple that behave like that,
and there are many
some involve mathematics, some simple models,
that produce behavior that don't even go that complicated,
Of course, not everything in looks very much like that.
Many things in the real world the are highly predictable.that.
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,
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,
And there are many things, periodic manner.
in the real world. of course, that are like that
And I'd like to show you another But not everything is.
That's the example of a planet example on the computer here.
Now this is one of the first orbiting a star.
modern sense in which we think problems ever solved in the
It was solved almost 400 years of solving a problem in science.
an astronomer, who was ago by Johannes Kepler,
Tycho Brahe. analyzing data from his mentor,
the planets orbit the sun in elliptical orbits.t
And he was able to deduce many
of those ellipses. of the mathematical properties
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,
thing is that we can predict or something, but the important
many years in advance. where the earth will be
predicting things in astronomy, we can even predict when an
eclipse of the sun will occur,
And we can even predict where on the earth to go, to stand,
to get the best view of that
And that's truly remarkable ethat the equations of physicsd.
known 400 years ago,
are sufficiently accurate and precise, and predictable,
that we can make those kinds
But my astronomy friends of long-term predictions.
throughout the universe, tell me that if you look
single stars, like our sun, but as binary stars, or even
groups of three or more stars.
when a planet orbits, You minot a single star,ppens
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
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.
It looks like it's doing And it looks almost random.
Now if you think about it, whatever it pleases.
because the underlying this is truly remarkable,
the same in these two problems. lawThey both involve lawsly
handed down to us by Isaac Newton 350 years ago.
They involve the universal law
are attracted with the force of gravitation, any two objects
to the square of the distance thatbetween them, and theynal
accelerate in proportion
So with those simple to that force.
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
very, very different. And this is an example,y
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.
this is a much more complicated Unlikkind of motion.lipse,
It may also surprise you to know
that I showed you, that the first problem
a planet plus a star, thewas solved 400 years ago,em,
but the three-body problem, which I'm showing you here,
has never been solved.
solved in the sense in which And probably never will be
in science. That is to say, there is nos
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.
to try to predict an eclipse And so it would be hopeless
or two ahead. inSo it's a very complicatedar
motion, and it is unpredictable.
And that's one of the characteristics
of chaotic motion.
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,
too far into the future, but if you try to go
Now another characteristic your predictions fail miserably.
it's complicated, and it doesn't of chaos, besides the fact that
has extreme sensitive dependence repeat, obviously, is that it
Now what does that mean. on initial conditions.
I started the planet right here When I sand I let it go,lation,
and it did its thing. You might wonder what would
happen if I would repeat
the planet in a little bit the calculation, but start
a velocity that's different by one part in a thousand.
Well, with the computer,
And that's what this shows. Here, I'm actually doing
two calculations and
So we have two planets here, but just overlaying the result.
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
somewhere. is here and one is way over here
means that you have to know So the initial conditionthis
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
the initial condition different rby one part in a million, by
this time, they would still be
And it's because the error doing something different.
grows exponentially. in the initial condition
know about exponential growth, Now the thing you probably all
it gets really big. is thatIt doesn't matter enough,
what you're talking about,
Do you have a question? >> You don't have an equation to
solve this, right?
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 equation, right? the computer is solving
be making the graph. Otherwise, I couldn't
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,
the time on the right hand side where yof the equation,ue for
and the left hand side
the planet's going to be. What the computer is doing,
is inching itself along,
because you can always predict in tiny little steps,
in the immediate future. what's going to happen
a long-term prediction, But without going throughmake
all of the intermediate steps.
that allow the error It's those intermediate steps
and to eventually to acompletely dominateply,
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?
extreme, wouldn't they, Tbecause at times you comey
very, very close to the stars,
you're much farther away. So that's a message
to the people who look
Don't bother looking around for binary star systems.fe.
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,
the ones that can be solved, and they've been solved long ago,
this is the best we can do. theBut there are many othered,
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
basically rely on some sort of periodic oscillation,
even our very best clocks,
rely on the oscillations atomic clocks,
kind of atom. So, it is some kind of regular
periodic oscillation that allows
And so it is in that sense that a periodic system, a regular
simply behaving system
is highly predictable. like that,
of that is this gadget Of course, another example
Those of you who are musicians right here.
an upside down pendulum. will recognize this as
It beats a very regular pattern. It's a metronome.
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,
far out, and that's bwhat the computer does.,
Do they have any, or you have any theory, or otherwise,
an idea of why that is so?
without any understanding of it? Is it just a fact,
of understanding. You're asking how do we know
when the behavior is going to be
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.
that are necessary for this Wekind of behavior, but evenns
when those conditions exist,
just looking at the equations that they're going to have
this kind of behavior,
on the computer and trying it. without actually putting it
this is an experimental science. So there is a sense in which
with trying to understand the conditions under which systems
exhibit this kind of behavior,
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,
But I can modify this pendulum highly predictable.
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,
from above, you will notice a very different kind of motion.
It's not simply swinging
around, not unlike what the baplanet does in the vicinityng
of the two stars. And in fact, those blue things,
as you might have suspected,
Two of them have their are magnets.
their north poles down. nAnd inside the tennis ball
is another magnet.
magnets and repelled by the It's attraother two.o of those
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.
of the characteristics. You have a sort of a tension
going on mathematically between two competing forces,
you might say.
of an answer to your question. And that's a little bit
you need to have for chaos. And if you just have two things
interacting, that is not
this kind of chaotic behavior. sufficient, usually, to produce
that there's something mystical Now yoand unusual aboutom this
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
a gravitational force. involved when there is only
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
in the equations. whYou can understand it,t
mathematically, in some sense.
there are many other examples of pendulums of various sorts.
Here's another pendulum.
at airport gift shops. This is a gadget you used to buy
for $8 about 20 years ago, I think I bought this one
And normally, you buy it I guess, now.
and you stare at it and youand day dream.r desk
But, it illustrates some important principles.
First of all, it looks like perpetual motion.
The thing's been going there
and it never seems to die. ever since I started talking,
thing as perpetual motion. But of course, there is no such
the bottom of that outer ring, There's a little magnet in
there's an electromagnet there and when it sweeps by the base,
and it gives it a little kick. that senses that it's coming by,
the swing. So it's like someone's pushing
back and forth in a regular periodic manner, approximately,
much like the pendulum
near it. before I put the magnets
inner ring, it's going around in a much more complicated
kind of fashion.
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,
for a couple of swings, thand then they would gether
out of step.
completely different things. Pretty soon, they would be doing
And, in fact, when I bought Sthis gadget, there was somes.
extra magnets on the inner ring and the outer ring, so when
they came by each other,
And I reasoned that those extra magnets weren't really essential
for the operation of it,
And in a great triumph and so I ripped them off.
the thing worked more or less of theothe same.hysics,
( light laughter )
because of, in this case, I suswe know the reason,d work
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.
maybe 20 degrees or so. It's Sorry, the outer ring.rth,
The inner ring, however,
And that's the critical element, occasionally goes over the top.
That brings into the mathematics that going over the top.
a non-linearity. something we call
It's a technical term, but it's one that comes up repeatedly in
those whole business of chaos.
about the linear case. First of all, let's talk
displace it away from its equilibrium down at the bottom,
there's a force pulling it back.
to the displacement. That force is proportional
If you displace it that way, That's what we mean by linear.
but the larger the displacement, the force brings it back,
and they are linearly related. the larger the force,
The inner ring, though, So twhen you start to go upans.
at a high elevation like this,
keep going up, up, up, up. the force doesn't
inverted, the force has gone In fact, when you get exactly
It's like a pendulum all the way to zero again.
There's no force at all there. balanced on its point.
non-linearity that is responsible for the chaos.
So you've just learned
to your question, in fact. something, in answer
for a system to exhibit chaos is that there be some kind of
non-linearity in the
So if you write down an equation mathematical equation.
without any thought, and it has no non-linearity,
Now that's not to say yothat every non-linear system.
exhibits chaos. In fact, the two-body problem
is non-linear, because
was inversely proportional remember the force
That's Newton's law of gravity. tAnd that's a non-linearity,e.
one over the square of something
So mathematicians say, is a non-linear force.
but not a sufficient condition non-linearity is a necessary,
So you've learned a bunch of things about chaos already.
You've learned
that it's complicated.
You've learned that it is tsensitively dependent.
on initial conditions.
it requires a non-linearity. And you've learned that
expert on this by this time. So you're getting to be pretty
unpredictable? >> By chaos, do you mean
in essence. >>That is one aspect of it.os,
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
But in the long-term, whatit's unpredictable.next.
And it's unpredictable because
on initial conditions. of this sensitive dependence
Yeah, back in the back. N>> The near-term predictions,--
which you say are accurate, are not really accurate,
they're only approximations.
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,
But it's never it's enormous.
Never completely accurate. completely accurate.
In fact, I can give you Well, ian example.n to be.
If I unplug the battery that's keeping this going,
I know the long-term solution to this.
( light laughter )
not moving. It's So it's not quite trueere,
that we can never predict.
we call transient chaos, There are examples of what
a chaotic behavior, wherebut it eventuallyhrough
settles down to something that is quite simple
and quite predictable.
of these things. And so, there are many examples
This is actually Now here's another pendulum.
It's a pendulum supported by another pendulum.
You're going to have trouble
I can hold it up a little bit. seeing this, aren't you?
And you can see a complicated Sokind of behavior,ms.
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.
what it's going to do. I know, eventually,
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.
this thing without friction, And if I could build
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
So let's see if I can do this fricwhile holding it. out.
I'm going to just swing the upper pendulum back and forth
in a regular manner.
is now swinging in a regular way And you see the upper pendulum
is now swinging ain a chaotic manner.m
So it's really the same example I showed over here.
The upper pendulum
to a very large amplitude. is not swinging
in a linear fashion, It's behaving
The lower pendulum can go all the way over the top,
and so it has a strong
to behave chaotically. non-linearity that causes it
of chaos in pendulums. So those are some examples
kind of pendulum. Here I have an upside down
U-shaped gadget.
One arm is free to swing It has two arms on it.
the other arm is rigid, and it also pivots in this way.
That's means there are basically
One kind of motion two kinds of motion.
this swinging back and forth, has this relike so. fixed and
There's another kind of motion
up and down and the whole thing where thisdoes that.e or less
But the two motions are obviously coupled together
in a fashion. And so if I start it up
like this, give it a lot
you see those motions cause it of initial energy,
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,
as close as I'm able to do. And you can well imagine that
they would stay together for
they would be doing a ltotally different things.ly,
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
and if there were no friction, of what we call transient chaos
It wouldn't be a transient, or if we somehow put energy
back into the system, that
So these are examples of chaos would keep it going forever.
Now this idea of chaos in mechanical systems.
to you as you might think. is really not as unfamiliar
we teach our physics students Oneis the way things fallngs
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 we teach them how to and it falls straight down.
of Newton, how long it takes calculate from the equations
it's going when it hits, to hit the ground, how fast
after I release it. wheAnd I can even do itant
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.
with that rubber ball, However, if instead of doing it
and drop it. I use this little index card
pretty straight. That time it dropped
You can see a very different Let's try it that way.
Now you can well imagine kind of behavior.
index cards, and I started them thatall as close as I couldthese
to the same condition and drop them,
there would be cards
And that's because the motion, all over the floor here.
it flutters in a way as it falls is chaotic,
to the exact starting condition. that is very sensitive
when you think about it. Now that's curious
exactly the same laws These twoof physics.e obeying
They're both being attracted
their gravity. to the earth by their weight,
air resistance, but yet the motion is somehow qualitatively
completely different
One is regular motion. in the two cases.
It may surprise you to know, One is chaotic motion.
ten years or so, that someone has actually written downt
the mathematical equation for that object falling
and demonstrating mathematically
and finding the conditions that indeed it should be chaotic
or becomes regular. uThere is a critical value ofc
the size, as you might imagine, and the shape of it that allows
it to behave chaotically. Now another thing
we teach students,
You know, physics we teis rocket science.kets.
And in the textbook, there are
of what happens when the exhaust all these nice pictures
of the rocket. comes out the back end
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,
but I do have a balloon. I don't have a rocket,
So of course, this is a rocket. There's some gas in there,
and when the gas
supposed to that way, and cthe book shows how that works.s
( laughter )
does it tell you about But nowhere in the textbook
At least not in the textbooks that we use to teach
our students at this university.
this is a chaotic motion. When it starts to leave,
it doesn't go in an absolutely straight line.
A slight departure
And in fact, that error in the starting position
grows exponentially.
repeated this with 100 balloons, Now you can imagine if I
exactly the same, there would be tried to start them all
up here, after a while. baI don't need to do theom
experiment for you
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.
we as educators, as physicists, in particular, have sort of
swindled our students,
has done this. and science in general
and maybe you share this belief, We have led students to believe,
that the essence of science as I dis predictability.ntly,
And if you can't predict what's going to happen, you really
aren't doing good science.
are situations, and Wthey're not at all uncommon,
where your best effort
is doomed to failure. You will never predict.
Another thing we think
to the scientific method is basic to science,
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
even in physics, of simple things in science,
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
or ten times and you get get different answers,
So it's a remarkable adjustment in our way of thinking.
A sort of a paradigm shift
science in this broader context. that we have to begin to see
we can't predict, and that That there are things
are, in fact, repeatable. notNow let's talk aboutnts
this pendulum a little bit. Yeah, you have another question?
When you say, starting out
exactly the same, I stop at the word "exactly."
Did I say exactly?
as far as we can make it the >> Does that mean "exactly"
Yes, I mis-spoke. same,When I said exactly,cal?
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
have different behavior. exactly the same, they would
besides the initial condition, Lots of things have to be exact,
the system itself, have to be the exactly identical,vern
or you will get this behavior.
and thanks for correcting me. ItOkay now, you may haven,
drawn the conclusion that chaos is complete disorder
and randomness. And I want to disabuse you
of that idea, because all of
in fact, can be described in these examples I've shown you,
that you can write down. terms of very simple equations
any randomness in them. AAnd you know at some levele
this cannot be a random process.
by what we call determinism, It is being governed
determined by the present. that the future is uniquely
peculiar characteristic that It's just that it has this
And so, it would be nice if we the initial conditions grow.
and convince ourselves that coulthis indeed is behavingthat,
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
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.
by an equation, you know But if you're deciding
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.
Let's go back Wellto the simple pendulum.that.
I tried to convince you that the pendulum, if it swings
through a small amplitude is periodic,
but if it swings
it is potentially chaotic. through a large amplitude,
imagine this pendulum swinging So in such a system, if you
being pushed regularly back and forth,
but the push is very regular to overcome the friction,
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
Okay, now imagine that we inlook 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,
where the inner pendulum is. we could ask
either experimentally, And we could do that,
on the computer. or we could simulate it
simulate that on the computer. And what I'm going to do is
periodically A pendulum being driven
Can you actually turn the lights by some external force.
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.
of equations that most It's a very simple set
could write down, except any freshman physics student
put them on the computer. to So here's a pendulumto
being driven back and forth.
The friction is being overcome It's got some friction.
So here we have a pendulum by this source of energy.
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
And if I do that, the pendulum just a little bit harder.
go to a large amplitude, will in fact,
go over the top. and even occasionally
what we mean by chaotic motion. You notice that the equations
that govern the system
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.
it is regular and repeatable. If And so, this is anotherugh,
characteristic of chaos.
with it a parameter, Chaos typically has associated
that you can adjust, and a knob, if you will,
you get regular behavior, for periodic in this case.ob,
For other values of that knob,
And so that's, I guess the fifth you get chaotic behavior.
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
And one thing you could do a little more than this.
Well, you could graph the angle is that the pendulum makesng.
versus time.
just swing back and forth, If it's periodic, it would
up and down, up and down. the Let's see what this oneown,
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
is moving. with which the pendulum
another complicated plot If you do that, you get
Now it could be that it repeats, that doesn't obviously repeat.
time, and you've forgotten but it just takes a very long
once you see it again. what it did before,
make a plot, on which the speed Sois on one axis, and theto
position is on the other axis.
computer has been calculating, And such a plot, which the
Now if this were look the something like this.
This would be an ellipse. It would simply be going around
and around in this funny space
and the angle of the pendulum. of the speed of the pendulum
what it did It in the previous cycle.ing
But here you see a much more
It looks like it can go comppretty much anywhereon.
it darn well pleases. And if you wait long enough,
it will be anywhere.
And you may wonder why a chaotic At least within these bounds.
bound here and here. system has such a rigid upper
this is not a random process. AndThis is a chaotic process.hat
It's obeying some rules.
is that you can never And yougo lower than thise rules
or higher than that.
of diagnosing a system to see that you're seeing evidence
of chaos. Now remember what it is.
This is a pendulum
And I'm plotting its speed going over the top.
at every instance of time. and its angle
at every instance of time, Suppose, instead of plotting
instances of time, like when I only plot at certain
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.
somewhere on these curves, Clearly those dots would be
fill in the whole space? but would that plot
because the computer, in fact, Well, let's look and see,
You can see, indeed, there are has been doing this.
do fall within those bounds lthat I showed you before,y
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.
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.
it attracts different And tinitial conditions.which
Now another thing you notice
sort of a complicated thing. about this, is that it's
this strange attractor In fact, this is an example,
mathematicians call a fractal. is an example of something
to talk about chaos without talking about fractals.
Fractals are sort of to chaos what geometry is to Algebra,
if you will.
the geometric manifestation of the chaotic dynamics.
And so if you plot something
chaotic, you will typically see in thsuch a fractal object.at is
Now there's a whole lecture I give on fractals.
It's a fascinating subject
give you the one-minute version. in its own right, but I'll just
object, like a plane A fractal is a mathematical
a peculiar characteristic of being self-similar.
Remember back in high school
about similar triangles? in geometry, you learned
They had the same angles, They wbut they wereshape.
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
What looks like a little zoom in on a piece of it here.
are really a bunch of lines, band here, you discover
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.
the structure you saw Andbefore you zoomed in.les
And so, you've learned something
but about a fractal. not about chaos,
another interesting property. NoIt has a dimensionas
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.
has zero dimensions. A point, or even a set of points
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
A surface, like a piece of with a single number.
is a two-dimensional object, paper to a good approximation,
object. orYou're used to the idea, andal
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.
that doesn't neatly fall into either of those categories.
This is somehow more than a very long line
that has been wrapped around
but it's less than a surface lots and lots times,
cut out of it. that has some holes
that is greater than one, It but smaller than two.act,
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
And there's a whole mathematics that's been built up around
fractal objects,
if this looks like something because it turns out
I'll come back to that you might have seen in nature,
there's no accident to that. in a moment,
of objects that have Because nature is full
They don't satisfy fractal characteristics.
exactly, but they do have the this characteristiction
that as you zoom in,
copies of it. youA tree is an example.ler
If you look at the limb of a tree, it looks a lot like
the trunk.
and those limbs have limbs, Andwhich have limbs,bs,
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.
self-similarity. It has a kind of a statistical
And I won't digress any more, except to say that looking for
fractals, in a dynamical system,
is an excellent way asof determining that you ares,
looking at a chaotic system,
Because a random system rtypically 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
of the non-chaotics thing that's driving it.
The periodic system that's driving it.
We could, of course, have made
of the driving system. a plot at various positions
is that if you're driving it And one thing you know
is that a succession in an irregular periodic manner,
a succession of such pictures, of such movies, or
to where it started. will eventually come back
kind of a movie. And so, you can make it
about the movie without doing it And one thing you know
endlessly. is that it's going to repeat
been collecting data for this. Well, the computer's
This is like 30 frames, I think. And here's what it's doing.
I'll see if I can make it It's go faster here.down.
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?
It looks like waves. Somebody said waves.
Whirlpool, yep. AnytSnail.lse?
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.
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
And I don't have some taffy. in a chaotic system.
right here, that I bought But down at the toy store.utty
If I work this like taffy,
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.
a kind of a horse shoe thing Now, you notice I've made
So imagine that I do that, when I stretch it and fold it.
If I do it a second time, you see the horse shoe
has now become
A little hard to hold it two smaller horse shoes.
and we have a bigger horse shoe. in my hand,
self-similarity comes about. So this is how all of that
I take the structure Every time I stretch and fold,
and I drive it down that to one smaller level. scale
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.
comes from. From the stretching and folding
of the so-called phase space. Now I can use this
to also illustrate
Remember these represent how this comes to be chaotic.
and velocities of a pendulum all the possible positions
I could put two marks on here. swinging back and forth.
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.
four times as far apart. I stretch it again, they're now
You can see where the error Eigin that starting conditiones.
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
Here's one of my favorites-- of other demonstrations.
I like to do things with balls. WeI showed you a tennis ball.e.
I showed you a super ball.
I'm going to put this ping pong Hball in a little petri dish.
And the petri dish is mounted on the cone of a loud speaker.
You don't actually
That's all you need to do. need to see this, just listen.
the speaker bounce up and down. WI've got it connected heree
to an audio oscillator
eight times a second. that Eight hertz.down
It's a little below what you can hear with your ear,
but you will be able to hear
Now you notice there's a knob. the ball bounce.
is the amplitude or the volume And the knob I'm adjusting
that's going into the speaker. of that eight hertz sound
is that under some Now what I want you to notice
it's a little hard to find adjustments of that knob,
For some values of that knob, it's like the metronome.
Maybe the table's a little unsteady,
that may be the problem.
the microphone closer. ( Let me put
Very regular and repeatable, rhythmic tapping sound )
makes a very different but a small change in this knob,
( random tapping sound ) kind of behavior.
of a chaotic system having a knob that you can turn,
and for some values of the knob,
For other values, it's completely chaotic.
And the transition
Now I want to show you another can be very abrupt.
And this is one of my favorite example of the same thing.
can all go home and repeat it. demonstrations, because you
any fancy equipment. You don't need
your kitchen or your bathroom, All you need is a faucet in
So here I have a beaker that yowith water in it.le bit.
And here's a faucet.
And when it drips on this, I have a little pie tin here.
when I put the microphone you'down next to it. it,
I'm going to open the faucet
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?
Let's see, what can I do? CouldnLet me drip itthat?
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 )
at night. It Well, if that happens,ke
here's an experiment you can do.
of the faucet. Go and adjust the drip rate
to make it drip faster Open it a little bit
Let's see if we can do that. and see what happens.
So here we have another system ( random dripping sound )
And for certain adjustments of that knob, it's very regular,
very predictable. For other adjustments
of the knob,
It seems to be going randomly. it seems to have no pattern.
it's an example of chaos. If you do this experiment at
home, here's something to try.
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.
it will go Instead of drip, drip, drip,
That's not very interesting. drip, drip, drip, drip.
point, and I won't try to do it But you wilhere.ch a certain
It's a little tedious.
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.
What will happen, is it should Turn it up a little bit more.
It sounds something like repeat every fourth drip.
drip, drip-drip-drip, drip, drip-drip-drip,
something like that.
a period doubling bifurcation. This is what's called
But you can see In twhat's happening.aos.
Instead of repeating every drip,
then every fourth, eighth, 16th. it repeats every other drip and
by which you have to turn Itthe knob is ever smallernt
when you go to higher and higher
And you eventually reach a point periods.
infinitely many drips whbefore it repeats.ly
And that's what we mean by chaos.
You've lost the periodicity.
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
Period four. it repeats.
period 32, finally chaos. That's a very nice experiment
of chaos that you can do
Now let's see, right in your home.
And that's chaos I have this one last example.
Many years ago, I actually in electrical circuits.
by being a ham radio operator. got interested in physics
I used to build all kinds of electronics.,
And so when I got to college
they assigned me to teach the electronics course.
I thought I was really sharp
I had been doing this since I was literally a baby almost.
And I even wrote a book,
And I never knew that there were "Modern Electronics."
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
you can make something something up in the right way,
being very complicated at all. exhibit chaos without it
electrical circuit here. And actually, I have a chaotic
the details of what's in it, I don't want to get into
it's a very, very simple but suffice its to say,
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.
for $600, they sell them for. AnThe money doesn't go to me.em
I get nothing.
You've all the experience Bof having a microphone
too close to the speaker
Audio feedback we call it, and you get feedback?
And this is one way to make a system chaotic, in fact.
Normally, when a system undergoes feedback,
you hear a screeching sound.
very loud, very annoying, A 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.
in a linear fashion, and that's Butwhy there is no chaos.ing
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 here is the tone so yothat it's making.ar it.
I'll hold it near the microphone
( monotone ) so you can all hear that.
was a tone being made with a microphone too closes
to a speaker and you're getting feedback to a particular
frequency.
And the knob in this case is basically the volume control.
As I turn up the volume control,
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
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
repeats, so it has two littles repeating, every other cycle
Big cycle, little cycle, and a big.
Period two. bigAnd then you havecle.
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
So that's the kind of feedback because chaos onsets.
circuit that had you would get in an audio
You could produce chaos the right kthat way.n-linearity.
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
we see complexity in the world, of losomething behaving when
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
Of course, we see in the world around us.
People who want to predict the there's a lot of interest.
would love to discover stock market and understand it,
equations you could write down that there's some simple model
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.
they see some animal species Or an ecologist who thinks that
this is terrible, there must be aboutsomething we're doing. oh,
It might not be anything we're doing.
It might just be inherent in the dynamics of the system
that things are going to
no matter what. Or the weather, we attribute a
lot of things to global warming these days.
You see fluctuations
and you say, oh, it must be of theglobal warming.winter
But maybe not, maybe there's
that if we could an underlying chaotic dynamic
with some simple system onof equations, we might learnt
quite a bit about it, without having to appeal
to all of the causes
that people imagine doing. and mechanisms
with a commercial. Now let me just close
you, every Tuesday noon If over in Chamberlin Hall,sts
we have a seminar,
"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,
but systems that exhibit not necessarily chaotic,
You're all welcome to come complex temporal dynamics.
There's only three or four more, and attend.
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
So thank you very much. ( applause )dance.
Can I take a question?
what kind of mathematics >> Yes, I'm just wondering
You can't use linear math. would describe this system.
they use to describe this? Is that abstract mathematics
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.
in the bank Iand it's earning interest.
And so, the bank is going to
by taking your current balance calculate your balance next year
Let's say you're earning and multiplying by some factor.
So, next year, you have 1.1 times what you have
this year.
and so forth. The yearMathematically,es that,
that's called iteration.
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,
It's a little like over and over and over.
You're wrapping it around that siland around.sn't it?
Or it's like the audio feedback.
and round, but in the process, Something's going round
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.
Your money is exponentially Yougrowing so that's nice,ver.
but very predictable
Now, something you all know is that something cannot grow
exponentially forever.
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.
as a society at some point, Maybeand our population this
is going to have to stop growing
It actually has already, exponentially.
The thing that stops the growth is typically a non-linearity.
Now the simplest non-linearity
is to say that, let's say yothe bank is going to do thisy
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.
by [1-x(n)]. But now we multiply it
compared to one in this case, Soit doesn't do very much.r
It's a linear growth. But as soon as X starts
to approach one,
So mathematically, this is this turns the growth rate down.
very realistic. an equAnything that growsething
exponentially, it eventually
And this system is capable stops it, puts a clamp on it.
That's all you need. of exhibiting chaos.
of A, that's about 3-1/2, You between 3-1/2 and 4.alue
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.
simple equations So yes, there are extremely
That was a great question. that exhibit this behavior.
So does the chaotic system Any others?
it's inclusive of linearity and regularity.
It's inclusive of,
Right. but a little greater than that.
has some parameter. In this case the A is them
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.
will almost always So indeed a chaotic system
it behaves non-chaotically. hAnd somewhat surprisingly,
even amidst the chaos,
there's lots of chaos, if you have a region where
a little bit, often, and find a window of periodic behavior,
for a small range of parameters.
there's an infinitely many In fact, for this equation,
region of the parameter A. such windows, even in a small
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.
should be able to do this. So any high school student
You can program this on It's the computer and seeebra.
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
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,
right under your feet you'and then to die off.aked
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
is going to give you a fractal. thatSome things do, plot
but many things don't.
a gaussian distribution by pooling together a lot
of observations,
That's right. Chaotic systems can mimic
a random process
and one way is that it can in many, many ways,
Thank you very much. ( make gaussian distributions.
applause )