
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

Not Synced
A nonlinear system is one where the
relationships between the variables that

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

Not Synced
at least in my car, where I fill up the
tank, and then I drive a hundred miles and

Not Synced
the needle barely moves. And then I drive
another hundred miles and the needle.

Not Synced
plummets. That's a nonlinear relationship
between the level of gas in the tank

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

Not Synced
Of all the systems in the universe that
evolves with time, that's the outer

Not Synced
ellipse in this Venn diagram, the vast
majority of them are nonlinear.

Not Synced
Indeed a famous mathematician refers to
the study of nonlinear dynamics as the

Not Synced
study of nonelephant animals. Now this is
somewhat problematic, because the

Not Synced
traditional training that we get in
science, engineering and mathematics uses

Not Synced
the assumption of linearity, and that's
only a very small part of the picture.

Not Synced
Now looking at the inner two ellipses on
this Venn diagram conveys the point that

Not Synced
the majority of nonlinear systems are
chaotic, and so that's gonna play a big

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

Not Synced
solved analytically, that is with a paper
and pencil, rather we have to solve them

Not Synced
with computers. And that is a large part
of what distinguishes this course on

Not Synced
nonlinear dynamics and chaos from most
other courses on this topic area,

Not Synced
including Steve Strogatz's great lectures
which are on the web, and the courses on

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

Not Synced
mathematics, but also on the role of
computation in the field. In this field,

Not Synced
the computer is the lab instrument. This
is experimental mathematics. And that's

Not Synced
actually why the field of nonlinear
dynamics only took off four decades ago

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

Not Synced
succeed in this course, you'll need to
understand the notion of a derivative,

Not Synced
because dynamical systems are about change
with time, and derivatives are the

Not Synced
mathematics of change with time. You'll
also need to be able to write simple

Not Synced
computer programs. Basically, to translate
simple mathematics formulas into code, run

Not Synced
them, and plot the results, say on the
axis of x versus t. There is no required

Not Synced
computer language. You can use
whichever programming language you want.

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

Not Synced
results that come out of it. You'll also
need to know about basic classical

Not Synced
mechanics, the stuff that you get in first
semester physics, like pendulums and

Not Synced
masses on springs, and bodies pulling on
each other, with GmM over rsquared kinds

Not Synced
of forces. Speaking of GmM over rsquared,
you may have seen this movie in the promo

Not Synced
video that I made. This is movie taken by
a camera on the Cassidy spacecraft as it

Not Synced
flew by Saturn's moon, Hyperion. Hyperion
is a very unusual shape and as a result of

Not Synced
that shape, it tumbles chaotically.
There's also chaos on how planets move

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

Not Synced
solutions in those cases can only be conic
sections, ellipses, parabolas and

Not Synced
hyperbolas. As we will see, systems with
three or more bodies can be chaotic. Now

Not Synced
think about it, how many bodies are there
in the solar system: lots more than two.

Not Synced
Indeed several hundred years, the King of
Sweden issued the challenge of a large

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

Not Synced
in the long term, and that prize was never
claimed. But the answer appeared in the

Not Synced
1980s. Indeed the solar system is chaotic,
although it is stable in a sense and we'll

Not Synced
get back to that. So just some brief
history of our field, it really dates back

Not Synced
to Henri Poincare in the late 1800s. But
it really got going in the 1960s with Ed

Not Synced
Lorentz's paper, called Deterministic Non
periodic Flow. Lorentz was the first

Not Synced
person to recognize the patterns of chaos
and the sensitivity of the evolution of

Not Synced
the system, within the context of those
patterns. In the 70s, this paper by Li and

Not Synced
Yorke was the first to use the word
"chaos" in conjunction with this behavior.

Not Synced
In the late 70s and 80s, the chaos cabal
at the University of California at Santa

Not Synced
Cruz, got very interested in nonlinear
dynamics, and one of the problems that

Not Synced
they approached it with was trying to beat
roulette, that is, modelling the path of a

Not Synced
ball on a roulette wheel, and using that
information to advantage. After this,

Not Synced
things really took off. And I should say,
of course, that I'm only cherrypicking a

Not Synced
very small number of examples by lots of
smart people in a very active field.

Not Synced
Nonlinear dynamics turns up all over the
place. Imagine an eddy in a creek, so a

Not Synced
patch of swirling water on the surface of
a creek or a river, you can imagine

Not Synced
dropping a wood chip in that patch of
water and watching its path from above,

Not Synced
perhaps with a camera, and then dropping
another wood chip in that eddy at a

Not Synced
slightly different point, and watching its
path. Those paths, they will trace out

Not Synced
the patches of swirling water in that eddy
in different order, but if you did a time

Not Synced
lapse photograph of their paths, they
would both trace out the same eddy.

Not Synced
Weather is nonlinear and chaotic. You may
have heard of the butterfly effect.

Not Synced
A butterfly flapping its wings setting off
a hurricane a week later, a thousand miles

Not Synced
away. Again, small change, large effect,
sensitive dependence on initial conditions

Not Synced
Marine invertebrates actually make use of
chaotic mixing in the water around them

Not Synced
during spawning, and I'm interested in
exploiting chaotic mixing to design better

Not Synced
fuel injectors in cars. Nonlinear and
chaotic dynamics also turns up in driven

Not Synced
nonlinear oscillators, like the pendulum
that I showed you, like the human heart

Not Synced
which is normally kind of mostly periodic
but, can go into a chaotic state called

Not Synced
ventricular fibrillation and as you saw
with the example of Hyperion, there's a

Not Synced
lot of nonlinear and chaotic dynamics in
classical mechanics ranging from the three

Not Synced
body problem and how black holes move
around each other. And nonlinear and

Not Synced
chaotic dynamics turns up in lots and lots
of other fields, including, certainly,

Not Synced
things that you are interested in. So as I
hope you can see, nonlinear and chaotic

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

Not Synced
and I hope that you will get infected by
some of that fascination over the course

Not Synced
of the next ten weeks. There are other
fascinating courses on the Complexity

Not Synced
Explorer website including Dave Feldman's
course on the same topic area that only

Not Synced
assumes knowledge of high school algebra,
and Melanie Mitchell's wonderful course on

Not Synced
complexity. The difference between
complexity and chaos actually bears a

Not Synced
little bit of explanation. Put perhaps too
simply, you can think of chaos as

Not Synced
complicated behavior from simple systems,
like my pendulum. And you can think of

Not Synced
complexity science as addressing systems
that are very complicated but have simple

Not Synced
behavior. Again, that is too pat but the
idea is generally right. So, a thousand

Not Synced
fish forming a single school. Now, some
logistics. There are several thousand of

Not Synced
you and one of me. We have an email
address for this course but it can very

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

Not Synced
the TA, for courserelated communications.
That thousandstoone ratios is one of the

Not Synced
major issues with MOOCs like this one.
Part of the way we plan to work around

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

Not Synced
staff, it's also to solve one of the other
problems with MOOCs, which is, instead of

Not Synced
being in a traditional classroom, everyone
taking this course is working by themselves

Not Synced
all over the world in all sorts of time
zones. And we hope to use the forum to

Not Synced
help with that. So if you've a question,
look on the forum. Someone else may have

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

Not Synced
answer, look at that answer. If you see a
question that you know the answer to, or

Not Synced
you think you do, offer your answer. I'll
also use the forum, by the way, to post

Not Synced
announcements, like there's a bug in the
problem set, or I've just posted a whole

Not Synced
new unit, or, the New York Times has an
article about the stuff I just talked about.

Not Synced
I'll also post discussion questions and
answers for topics that may interest some

Not Synced
people in the course, if somebody wants to
go deeper into something or sideways along

Not Synced
a tangent, that's where the forum can
play a role.

Not Synced
Here's another piece of technology that
can help.

Not Synced
There's no textbooks for this course. I'm
pulling together material from many many

Not Synced
different sources, including a substantial
amount from my own work, papers that I've

Not Synced
read, talks that I've heard at conferences
and so on and so forth.

Not Synced
These video lectures are short, self
contained summaries of each topic. I use

Not Synced
the Supplementary Materials page to
supplement those summaries. So if you want

Not Synced
to dig more deeply into something I
mentioned, or you'd like some background

Not Synced
material, or, you wanna read the original
paper that I mentioned. This is where you

Not Synced
should look. In the next segment of this
course, we'll start digging into some

Not Synced
ideas and mathematics and plots and
computer examples. Most of my video

Not Synced
lectures, by the way, are not quite as long
as this one. We had a lot to cover today.

Not Synced
And there will be a short quiz after most
of my video lectures, a way for you to

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

Not Synced
end of each unit, of which there are ten,
there will be a unit test. Those are

Not Synced
graded electronically, and that grade will
be the basis of your eligibility for a

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

Not Synced
a certificate. You may just wanna watch
the lectures, and that's absolutely fine.

Not Synced
This is all here on offer for you to use
in the way that best suits you.

Not Synced
A word about computers. Functional
computer literacy is a prerequisite for

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

Not Synced
that you will need to explore in the
homework. Now, I've designed the course so

Not Synced
that you can still pass it without doing
that and you can still get a flavor of the

Not Synced
concepts. But to get the full experience,
you really do need to be able to do the

Not Synced
homework. And there will be problems on
each exam that depend on your having done

Not Synced
the programming for the homework for that
unit. You're welcome to use any computer

Not Synced
programming language that you wish, modern
computer programming languages are all

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

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

Not Synced
We're interested in what comes out and
that's what we'll be looking for in the

Not Synced
exams and the quizzes. Another related and
important point, there are thousands of

Not Synced
you, and among the thousands of you, there
are going to be dozens of favorite

Not Synced
programming languages, so there's no way
that we'll be able to help you debug your

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

Not Synced
just post entire solutions on the forum
and ask, "Where's the bug?" We have

Not Synced
chosen Matlab as the program in which we
will post our solutions, because it's

Not Synced
pretty widespread and pretty simple. It's
a good lingua franca for that purpose. If

Not Synced
you've never encountered Matlab, you may
want to look over one of the many

Not Synced
tutorials that are available on the web
for the basic syntax for that language so

Not Synced
that you can understand our solutions.