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

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

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

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

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

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

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

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Moreover, this system is sensitively
dependent on dynamical systems. If I

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

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

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

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

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

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

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the current state determines the future
state.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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complexity. The difference between
complexity and chaos actually bears a

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Here's another piece of technology that
can help.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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A word about computers. Functional
computer literacy is a