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← Nonlinear 2.3 Exploring the bifurcation diagram

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Showing Revision 2 created 12/16/2016 by Miharu Jay Kimwell.

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    In the last segment, I showed you how
    return maps work,
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    how to go back and forth between
    them and the time domain,
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    and how they help you understand the
    dynamics, as well as
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    how to understand bifurcations in the
    dynamics as the parameter value changes.
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    I finished up with a 3rd representation,
    the bifurcation diagram.
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    Here's a bifurcation diagram of
    the logistic map.
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    On the vertical axis is a set of iterates
    of the logistic map,
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    at some parameter value R, which is
    graphed on the horizontal axis.
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    Just to remind you of the correspondence
    between this kind of plot
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    and the time domain, and the return map,
    I'm gonna draw a few pictures.
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    Here's a time domain plot of an orbit of
    the logistic map
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    at a low value of the parameter R
    that is converging to a fixed point.
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    On a return map, this orbit would look
    like this.
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    To construct a bifurcation diagram,
    you remove the transient;
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    that is, you iterate a bunch of times,
    and throw those points away,
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    and then you iterate a bunch more times,
    and you plot those points
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    as if you were looking at the top plot
    edge-on from the side.
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    In this case, those points would all fall
    on top of each other, there.
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    So again, each vertical slice of the
    bifrucation diagram
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    is one time-domain plot like this, with
    the transient removed,
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    viewed from the side.
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    If we turn R up a little bit, the time-
    domain plot will look like this,
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    the return map will look like this,
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    and the point on the bifurcation diagram
    will look like 2 dots.
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    Again, three different representations
    bring out three different things:
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    the time-domain plot on the top left
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    brings out the overall behavior of
    the iterates;
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    the return map on the lower left
    brings out the geometry of why
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    the iterates go where they go, & also the
    correlation between successive iterates;
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    the bifurcation plot brings out what
    changes about the asymptotic behavior
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    of the trajectory as R changes,
    including bifurcations.
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    Now if you repeat the procedure that we
    just went through at a much finer grain,
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    but using a computer instead of a tablet
    and a stylus, what you'll see is this.
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    There's actually one more step in there
    which we'll circle back to
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    at the end of this segment.
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    Now, you can see all sorts of structure
    in this plot.
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    That's the main focus of this segment.
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    First of all, you see the fixed point
    coming along for low R, here,
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    and then bifurcating into a 2-cycle
    right here,
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    bifurcating into a 4-cycle right here,
    and then eventually,
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    getting into a chaotic regime. That's
    what this gray banded behavior is.
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    That's what this right-hand plot would
    look like if you looked at it edge-on
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    from the right-hand side of the screen.
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    Within the chaotic regimes, you also see
    these "veils":
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    areas where the attractor is darker
    than in other areas.
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    Those veils are related to what are called
    "unstable periodic orbits",
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    and we'll talk more about them later.
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    As we've seen, there's this bifurcation
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    from a fixed point, to a 2-cycle, to a 4-
    cycle, to an 8-cycle, & so on & so forth.
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    That's called a "period-doubling cascade"
    for the obvious reason.
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    I also showed you in the last segment
    that there were regions of order
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    within the chaos; that is, for some
    R-value, there was chaos,
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    but then if you raised R a little bit, you
    went back into a periodic regime.
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    This particular periodic regime starts out
    with a 3-cycle, and then goes to a 6-cycle,
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    and a 12-cycle, and so on and so forth.
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    So it's another periodic doubling
    bifurcation sequence.
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    You may remember, in the very first
    segment of this course,
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    I showed you the title page of a paper
    called "Period-3 Implies Chaos".
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    The fact there is a period-3 orbit
    in this map is very, very significant.
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    And if people are interested in that, I
    can record an auxiliary video about that.
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    Another interesting thing to note about
    this structure
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    is that it contains small copies of itself.
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    If you were to zoom in on that piece of
    the structure inside the red circle,
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    it would look like the whole structure.
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    That is, this is a fractal object.
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    I'm sure many of you have hear about fractals.
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    Fractals are sets that have non-integer
    Hausdorff dimension
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    (mathematically, that's the formal term).
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    Informally, they're "self-similar".
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    The second row of images here show
    something called the Koch curve.
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    The way you construct this triangle is by
    taking an equilateral triangle,
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    and then taking 3 equilateral triangles,
    1/3 the size in the sense of edge-length,
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    and sticking them to each exposed face
    of that thing.
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    Then you iterate; you take little
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    and stick them to the sides of each of
    those pointy faces, and keep going.
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    Eventually, you'll get this beautiful
    structure that looks alot like a snowflake
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    Fractals play an interesting role
    in mathematics.
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    There are lots of examples of fractals
    and fractal-like structures in nature.
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    Here's an example.
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    Fractals are also useful analogs
    for nature in computer graphics.
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    Here's a beautiful fractal called
    the Mandelbrot set,
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    and this video is showing you that if you
    zoom in on the Mandelbrot set,
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    you keep seeing more and more structure;
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    in fact, you keep seeing structure that is
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    There's a whole new Mandelbrot set
    way down in the tendrils of the old one.
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    And you can keep zooming in
    and zooming in,
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    and you'll keep seeing
    self-similar structure.
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    I've included a link to that video on the
    supplementary materials section
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    of the Complexity Explorer website
    for this course,
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    right here, under the section for this
    segment of this unit.
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    Remember, this is where you should go
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    for links to materials that you might need
    to do the homework,
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    like this Logistic Map app,
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    for materials like this paper, which you
    would look at
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    if you wanted to learn more