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 non-elephant 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 r-squared kinds
of forces. Speaking of GmM over r-squared, 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 cherry-picking 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 course-related communications. That thousands-to-one 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.