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← Nonlinear 2.3 Exploring the bifurcation diagram

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Showing Revision 4 created 12/16/2016 by Miharu Jay Kimwell.

  1. In the last segment, I showed you how
    return maps work,
  2. how to go back and forth between
    them and the time domain,
  3. and how they help you understand the
    dynamics, as well as
  4. how to understand bifurcations in the
    dynamics as the parameter value changes.
  5. I finished up with a 3rd representation,
    the bifurcation diagram.
  6. Here's a bifurcation diagram of
    the logistic map.
  7. On the vertical axis is a set of iterates
    of the logistic map,
  8. at some parameter value R, which is
    graphed on the horizontal axis.
  9. Just to remind you of the correspondence
    between this kind of plot
  10. and the time domain, and the return map,
    I'm gonna draw a few pictures.
  11. Here's a time domain plot of an orbit of
    the logistic map
  12. at a low value of the parameter R
    that is converging to a fixed point.
  13. On a return map, this orbit would look
    like this.
  14. To construct a bifurcation diagram,
    you remove the transient;
  15. that is, you iterate a bunch of times,
    and throw those points away,
  16. and then you iterate a bunch more times,
    and you plot those points
  17. as if you were looking at that top plot
    edge-on from the side.
  18. In this case, those points would all fall
    on top of each other, there.
  19. So again, each vertical slice of the
    bifrucation diagram
  20. is one time-domain plot like this, with
    the transient removed,
  21. viewed from the side.
  22. If we turn R up a little bit, the time-
    domain plot will look like this,
  23. the return map will look like this,
  24. and the point on the bifurcation diagram
    will look like 2 dots.
  25. Again, three different representations
    bring out three different things:
  26. the time-domain plot on the top left
  27. brings out the overall behavior of
    the iterates;
  28. the return map on the lower left
    brings out the geometry of why
  29. the iterates go where they go, & also the
    correlation between successive iterates;
  30. the bifurcation plot brings out what
    changes about the asymptotic behavior
  31. of the trajectory as R changes,
    including bifurcations.
  32. Now if you repeat the procedure that we
    just went through at a much finer grain,
  33. but using a computer instead of tablet
    and a stylus, what you'll see is this.
  34. There's actually one more step in there
    which we'll circle back to
  35. at the end of this segment.
  36. Now, you can see all sorts of structure
    in this plot.
  37. That's the main focus of this segment.
  38. First of all, you see the fixed point
    coming along for low R, here,
  39. and then bifurcating into a 2-cycle
    right here,
  40. bifurcating into a 4-cycle right here,
    and then eventually,
  41. getting into a chaotic regime. That's
    what this gray banded behavior is.
  42. That's what this right-hand plot would
    look like if you looked at it edge-on
  43. from the right-hand side of the screen.
  44. Within the chaotic regimes, you also see
    these "veils":
  45. areas where the attractor is darker
    than in other areas.
  46. Those veils are related to what are called
    "unstable periodic orbits",
  47. and we'll talk more about them later.
  48. As we've seen, there's this bifurcation
    sequence
  49. from a fixed point, to a 2-cycle, to a 4-
    cycle, to an 8-cycle, & so on & so forth.
  50. That's called a "period-doubling cascade"
    for the obvious reason.
  51. I also showed you in the last segment
    that there were regions of order
  52. within the chaos; that is, for some
    R-value, there was chaos,
  53. but then if you raised R a little bit, you
    went back into a periodic regime.
  54. This particular periodic regime starts out
    with a 3-cycle, and then goes to a 6-cycle
  55. and a 12-cycle, and so on and so forth.
  56. So it's another period-doubling
    bifurcation sequence.
  57. You may remember, in the very first
    segment of this course,
  58. I showed you the title page of a paper
    called "Period-3 Implies Chaos".
  59. The fact there is a period-3 orbit
    in this map is very, very significant.
  60. And if people are interested in that, I
    can record an auxiliary video about that.
  61. Another interesting thing to note about
    this structure
  62. is that it contains small
    copies of itself.
  63. If you were to zoom in on that piece of
    the structure inside the red circle,
  64. it would look like the whole structure.
  65. That is, this is a fractal object.
  66. I'm sure many of you have heard about fractals.
  67. Fractals are sets that have non-integer
    Hausdorff dimension
  68. (mathematically, that's the formal term).
  69. Informally, they're "self-similar".
  70. The second row of images here show
    something called the Koch curve.
  71. The way you construct this fractal is by
    taking an equilateral triangle,
  72. and then taking 3 equilateral triangles,
    1/3 the size in the sense of edge-length,
  73. and sticking them to each exposed face
    of that thing.
  74. Then you iterate; you take little
    triangles
  75. and stick them to the sides of each of
    those pointy faces, and keep going.
  76. Eventually, you'll get this beautiful
    structure that looks alot like a snowflake
  77. Fractals play an interesting role
    in mathematics.
  78. There are also lots of examples of frac-
    tals & fractal-like structures in nature.
  79. Here's an example.
  80. Fractals are also useful analogs
    for nature in computer graphics.
  81. Here's a beautiful fractal called
    the Mandelbrot set,
  82. and this video is showing you that if you
    zoom in on the Mandelbrot set,
  83. you keep seeing more and more structure;
  84. in fact, you keep seeing structure that is
    self-similar.
  85. There's a whole new Mandelbrot set
    way down in the tendrils of the old one.
  86. And you can keep zooming in
    and zooming in,
  87. and you'll keep seeing
    self-similar structure.
  88. I've included a link to that video on the
    supplementary materials section
  89. of the Complexity Explorer website
    for this course,
  90. right here, under the section for this
    segment of this unit.
  91. Remember, this is where you should go
  92. for links to materials that you might need
    to do the homework,
  93. like this Logistic Map app,
  94. for materials like this paper, which you
    would look at
  95. if you wanted to learn more about the
    concepts that I talked about
  96. in that segment.
  97. And I've also included some links to
    tutorial materials
  98. and other sorts of things that might help
    you if you need some background to fill in
  99. And here's an important thing: the
    connection between fractals and chaos.
  100. There is a connection, but it is not an
    "if-and-only-if".
  101. Many chaotic systems have some
    fractal structure,
  102. but it is by no means the case that all
    chaotic systems have fractal structure;
  103. that is, there are chaotic systems that
    do not have fractal structure,
  104. there are certainly tons of fractals
    that have nothing to do with chaos,
  105. but the popular science press has
    conflated these two topics.
  106. If you want to learn more about fractals,
  107. you can take a look at Dave Feldman's
    course on the Complexity Explorer MOOC.
  108. One last point here, relating to
    transient length:
  109. remember that for some R-values,
    the transient was really long?
  110. How do you think that will manifest
    in a bifurcation diagram?
  111. That is, there is some fixed point here,
    but the trajectory is taking
  112. a really long time to get there.
  113. What that will look like on a slice of the
    bifurcation diagram is this.
  114. That's hard to see, but I'm trying to draw
    a series of points coming up from the axis
  115. and slowly getting closer and closer and
    closer, but taking forever to get there.
  116. So if we want to see the asymptotic
    behavior,
  117. we want to throw out the transient, but
    how many points do we need to throw out
  118. if we want to get rid
    of the transient here?
  119. To get rid of the transient, we actually
    need another step in our code here.
  120. Really what we need to do is iterate a
    whole bunch of times,
  121. but not plot those points,
  122. and then from the ending point of
    that orbit,
  123. iterate a bunch more times, and plot
    those points.
  124. That amounts to omitting the transient.
  125. But the question is, these words: how do
    you pick how many points to iterate
  126. to get rid of the transient, and how do
    you pick how many points to plot
  127. so that you get a really nice picture?
    Those are both tricky.
  128. You want the red bunch number to be
    large enough so that you see the structure
  129. but not so large that the finite size of
    the plotted points obscures the structure.
  130. And you want to throw out enough points
    so the transient has really died out,
  131. but how long is that? There's no way to
    know, really.
  132. And they tend to get longer just before
    a bifurcation.
  133. In practice, what you do is increase the
    number of points that you throw away
  134. before plotting, until the periodic orbits
    are crisp on your plots.
  135. That amount of thrown-away points is
    overkill far away from the bifurcations,
  136. of course, where the transient is short,
  137. but otherwise, your orbits will thicken
    up near the bifurcation point.
  138. All of that will play a role in the next
    segment, where we'll dig into the pattern
  139. behind the shrinking widths and heights
    of the pitchforks in the bifurcation plot.