Hi this is Liz Bradley, I'm a Professor
in the Computer Science department
in the University of Colorado at Boulder
and also on the external faculty of the
Santa Fe Institute. My research interests
are on nonlinear dynamics and chaos and
in artificial intelligence, and I'm going
to be your guide in 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 very simple, it's
behavior is very complicated.
Moreover, this system is sensitively
dependent on dynamical systems. 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
evolves 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 a 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 off 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