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← 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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    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 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