← Nonlinear 1.1 Introduction to nonlinear dynamics

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Showing Revision 8 created 07/20/2016 by Steven Gunawan.

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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 non-elephant 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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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 r-squared kinds
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of forces. Speaking of GmM over r-squared,
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