
Hi this is Liz Bradley, I'm a Professor
in the Computer Science department

at the University of Colorado at Boulder
and also on the external faculty of the

Santa Fe Institute. My research interests
are in nonlinear dynamics and chaos and

in artificial intelligence, and I'm going
to be your guide during this course on

nonlinear dynamics and chaos. Here's an
example of a nonlinear dynamical system.

It's a double pendulum. Two pieces of
aluminium and four ball bearings. Even

though the system is physically very
simple, it's behavior is very complicated.

Moreover, this system is sensitively
dependent on initial conditions. If I

started here, or here, the future evolution
of the behavior will be very different.

Even though the behavior of that device is
very very complicated, there are some very

strong patterns in that behavior, and the
tandem of those patterns and the sensitivity

is the hallmark of chaos. Now there's
lots of words on this slide that we'll get

into over the next ten weeks. I'll just
give you some highlights here.

A deterministic system is one that is not
random. Cause and effect are linked and

the current state determines the future
state.

A dynamic system (or a dynamical system),
either are fine, is a system that evolves with time

A nonlinear system is one where the
relationships between the variables that

matter are not linear. An example of a non
linear system is the gas gauge in a car,

at least in my car, where I fill up the
tank, and then I drive a hundred miles and

the needle barely moves. And then I drive
another hundred miles and the needle

plummets. That's a nonlinear relationship
between the level of gas in the tank

and the position of the needle. Now non
linear dynamics and chaos are not rare.

Of all the systems in the universe that
evolve with time, that's the outer

ellipse in this Venn diagram, the vast
majority of them are nonlinear.

Indeed a famous mathematician refers to
the study of nonlinear dynamics as the

study of nonelephant animals. Now this is
somewhat problematic, because the

traditional training that we get in
science, engineering and mathematics uses

the assumption of linearity, and that's
only a very small part of the picture.

Now looking at the inner two ellipses on
this Venn diagram conveys the point that

the majority of nonlinear systems are
chaotic, and so that's gonna play a big

role in this course. And the equations
that describe chaotic systems cannot be

solved analytically, that is with paper
and pencil, rather we have to solve them

with computers. And that is a large part
of what distinguishes this course on

nonlinear dynamics and chaos from most
other courses on this topic area,

including Steve Strogatz's great lectures
which are on the web, and the courses on

the complexity explorer website about this
topic. We will focus not only on the

mathematics, but also on the role of
computation in the field. In this field,

the computer is the lab instrument. This
is experimental mathematics. And that's

actually why the field of nonlinear dynamics
only took of three or four decades ago

Before that, there weren't computers to
help us solve the equations. Now to

succeed in this course, you'll need to
understand the notion of a derivative,

because dynamical systems are about change
with time, and derivatives are the

mathematics of change with time. You'll
also need to be able to write simple

computer programs. Basically, to translate
simple mathematics formulas into code, run

them, and plot the results, say on the
axis of x versus t. There is no required

computer language. You can use
whichever programming language you want.

And you're not gonna turn in your code in
this course. We're interested in the

results that come out of it. You'll also
need to know about basic classical

mechanics, the stuff that you get in first
semester physics, like pendulums and

masses on springs, and bodies pulling on
each other, with GmM over rsquared kinds

of forces. Speaking of GmM over rsquared,
you may have seen this movie in the promo

video that I made. This is movie taken by
a camera on the Cassidy spacecraft as it

flew by Saturn's moon, Hyperion. Hyperion
is a very unusual shape and as a result of

that shape, it tumbles chaotically.
There's also chaos on how planets move

through space, not just how they tumble.
You may remember from Physics, that the

solutions in those cases can only be conic
sections, ellipses, parabolas and

hyperbolas. As we will see, systems with
three or more bodies can be chaotic. Now

think about it, how many bodies are there
in the solar system: lots more than two.

Indeed several hundred years, the King of
Sweden issued the challenge of a large

cash prize to the person who could prove
whether or not the solar system was stable

in the long term, and that prize was never
claimed. But the answer appeared in the

1980s. Indeed the solar system is chaotic,
although it is stable in a sense and we'll

get back to that. So just some brief
history of our field, it really dates back

to Henri Poincare in the late 1800s. But
it really got going in the 1960s with Ed

Lorentz's paper, called Deterministic Non
periodic Flow. Lorentz was the first

person to recognize the patterns of chaos
and the sensitivity of the evolution of

the system, within the context of those
patterns. In the 70s, this paper by Li and

Yorke was the first to use the word
"chaos" in conjunction with this behavior.

In the late 70s and 80s, the chaos cabal
at the University of California at Santa

Cruz, got very interested in nonlinear
dynamics, and one of the problems that

they approached it with was trying to beat
roulette, that is, modelling the path of a

ball on a roulette wheel, and using that
information to advantage. After this,

things really took off. And I should say,
of course, that I'm only cherrypicking a

very small number of examples by lots of
smart people in a very active field.

Nonlinear dynamics turns up all over the
place. Imagine an eddy in a creek, so a

patch of swirling water on the surface of
a creek or a river, you can imagine

dropping a wood chip in that patch of
water and watching its path from above,

perhaps with a camera, and then dropping
another wood chip in that eddy at a

slightly different point, and watching its
path. Those paths, they will trace out

the patches of swirling water in that eddy
in different order, but if you did a time

lapse photograph of their paths, they
would both trace out the same eddy.

Weather is nonlinear and chaotic. You may
have heard of the butterfly effect.

A butterfly flapping its wings setting off
a hurricane a week later, a thousand miles

away. Again, small change, large effect,
sensitive dependence on initial conditions

Marine invertebrates actually make use of
chaotic mixing in the water around them

during spawning, and I'm interested in
exploiting chaotic mixing to design better

fuel injectors in cars. Nonlinear and
chaotic dynamics also turns up in driven

nonlinear oscillators, like the pendulum
that I showed you, like the human heart

which is normally kind of mostly periodic
but, can go into a chaotic state called

ventricular fibrillation and as you saw
with the example of Hyperion, there's a

lot of nonlinear and chaotic dynamics in
classical mechanics ranging from the three

body problem to how black holes move
around each other. And nonlinear and

chaotic dynamics turns up in lots and lots
of other fields, including, certainly,

things that you are interested in. So as I
hope you can see, nonlinear and chaotic

dynamics are not an academic oddity. They
are widespread, and they are fascinating,

and I hope that you will get infected by
some of that fascination over the course

of the next ten weeks. There are other
fascinating courses on the Complexity

Explorer website including Dave Feldman's
course on the same topic area that only

assumes knowledge of high school algebra,
and Melanie Mitchell's wonderful course on

complexity. The difference between
complexity and chaos actually bears a

little bit of explanation. Put perhaps too
simply, you can think of chaos as

complicated behavior from simple systems,
like my pendulum. And you can think of

complexity science as addressing systems
that are very complicated but have simple

behavior. Again, that is too pat but the
idea is generally right. So, a thousand

fish forming a single school. Now, some
logistics. There are several thousand of

you and one of me. We have an email
address for this course but it can very

rapidly get overwhelmed. Please do not use
my own personal email address, or that of

the TA, for courserelated communications.
That thousandstoone ratios is one of the

major issues with MOOCs like this one.
Part of the way we plan to work around

that is with an electronic forum. This is
not just to take a load off the course

staff, it's also to solve one of the other
problems with MOOCs, which is, instead of

being in a traditional classroom, everyone
taking this course is working by themselves

all over the world in all sorts of time
zones. And we hope to use the forum to

help with that. So if you've a question,
look on the forum. Someone else may have

posted that question already. If not,
post it yourself. If someone has posted an

answer, look at that answer. If you see a
question that you know the answer to, or

you think you do, offer your answer. I'll
also use the forum, by the way, to post

announcements, like there's a bug in the
problem set, or I've just posted a whole

new unit, or, the New York Times has an
article about the stuff I just talked about.

I'll also post discussion questions and
answers for topics that may interest some

people in the course, if somebody wants to
go deeper into something or sideways along

a tangent, that's where the forum can
play a role.

Here's another piece of technology that
can help.

There's no textbooks for this course. I'm
pulling together material from many many

different sources, including a substantial
amount from my own work, papers that I've

read, talks that I've heard at conferences
and so on and so forth.

These video lectures are short, self
contained summaries of each topic. I use

the Supplementary Materials page to
supplement those summaries. So if you want

to dig more deeply into something I
mentioned, or you'd like some background

material, or, you wanna read the original
paper that I mentioned. This is where you

should look. In the next segment of this
course, we'll start digging into some

ideas and mathematics and plots and
computer examples. Most of my video

lectures, by the way, will not be quite as long
as this one. We had a lot to cover today.

And there will be a short quiz after most
of my video lectures, a way for you to

rote test your understanding of the
material. Those will not be graded. At the

end of each unit, of which there are ten,
there will be a unit test. Those are

graded electronically, and that grade will
be the basis of your eligibility for a

certificate of completion of this course,
if you want one. Some of you may not want

a certificate. You may just wanna watch
the lectures, and that's absolutely fine.

This is all here on offer for you to use
in the way that best suits you.

A word about computers. Functional
computer literacy is a prerequisite for

this course. If you can't program, you're
not gonna be able to write the programs

that you will need to explore in the
homework. Now, I've designed the course so

that you can still pass it without doing
that and you can still get a flavor of the

concepts. But to get the full experience,
you really do need to be able to do the

homework. And there will be problems on
each exam that depend on your having done

the programming for the homework for that
unit. You're welcome to use any computer

programming language that you wish, modern
computer programming languages are all

Turing equivalents, so it shouldn't matter
what you use. What's gonna matter is what

comes out of your code, not the how well
commented it is or what style it has.

We're interested in what comes out and
that's what we'll be looking for in the

exams and the quizzes. Another related and
important point, there are thousands of

you, and among the thousands of you, there
are going to be dozens of favorite

programming languages, so there's no way
that we'll be able to help you debug your

code. You can post on the forum, and your
classmates will help you. Please do not

just post entire solutions on the forum
and ask, "Where's the bug?" We have

chosen Matlab as the program in which we
will post our solutions, because it's

pretty widespread and pretty simple. It's
a good lingua franca for that purpose. If

you've never encountered Matlab, you may
want to look over one of the many

tutorials that are available on the web
for the basic syntax for that language so

that you can understand our solutions.