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

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in 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 that
chaos. 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, Santa

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Cruz, got very interested