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Showing Revision 2 created 06/10/2014 by JULIE BEAL.

  1. but first just a little bit of terminology
  2. which should be familiar
  3. from what we did with differential equations
  4. I would say the system undergoes a bifurcation here
  5. at r = 3
  6. remember a bifurcation is a sudden
  7. qualitative change in the behaviour of a
  8. dynamical system
  9. as a parameter is varied continuously
  10. so the qualitative change here
  11. is that the fixed point here splits into two
  12. so we go from an attractor of period 1
  13. to an attractor of period 2
  14. so that's a bifurcation
  15. and it's called a period doubling bifurcation
  16. because the period doubles
  17. here we see we have a bifurcation
  18. from period 2 to period 4
  19. so that's another period doubling bifurcation
  20. ok, let's zoom in on the bifurcation diagram
  21. let's look at just this portion
  22. let's look at what's going on from 3 to 4
  23. since this is where a lot of the interesting action is
  24. so here, I've zoomed in,
  25. and this is a bifurcation diagram from 3 to 4
  26. So we see in this region, from 3 to about a little more than 3.4,
  27. the behavior is period 2
  28. Here are the final state diagrams we drew previously.
  29. There's 3.2, and there's 3.4,
  30. and they line up pretty well.
  31. Let's see if I can get a few more on here.
  32. Here's 3.739, and that corresponds to
  33. this funny region here
  34. (this light region)
  35. and we'll look at that more closely in a bit.
  36. But period 5 (1, 2, 3, 4, 5)
  37. and then we had two 8 periodic values,
  38. at 3.8 ... at around there,
  39. that looks pretty good
  40. and then 3.9 which is right around there.
  41. So, the bifurcation diagram for the logistic map
  42. looks quite different than the one we saw
  43. for differential equations,
  44. which isn't surprising,
  45. the logistic map, and things like it,
  46. exhibit chaos, aperiodic behavior,
  47. so we'd expect it to be a richer bifurcation diagram,
  48. and have more features to look at.
  49. But remember, the thing about bifurcation diagrams,
  50. to interpret them,
  51. remember that they began their life (in this case)
  52. as a series of final state diagrams
  53. so for example, if I wanted to know
  54. what's going on right about 3.7,
  55. I would just try to blot out everything except for 3.7,
  56. and then view it as a single final state diagram.
  57. I sort of imagined doing that with this thing that I've made.
  58. So ... I've moved this so that the slit shows right around 3.7,
  59. and so we would say that this looks like an aperiodic region,
  60. lots and lots of dots,
  61. so it must be aperiodic,
  62. going between this value and this value.
  63. If I wanted to know what' going on at 3.2,
  64. I could move this until I'm seeing 3.2,
  65. and then I would see just these two dots here,
  66. (or small line segments)
  67. and that would mean that this is periodic with period 2.
  68. And imagine another way to view this,
  69. as r increases, we see period 2 behavior,
  70. and the two values are getting farther apart.
  71. They're moving this way, as I let r get larger.
  72. And then a little past 3.4,
  73. (where is it? there it is - there's a bifurcation)
  74. so now it's period 4 (1, 2, 3, 4).
  75. A small change in r (that's moving this)
  76. leads to a qualitative change
  77. in the behavior of the dynamical system.
  78. In this case we go from a cycle of period 2 to a cycle of period 4
  79. and then as I increase r further still,
  80. there's a region of period 8 (1, 2, 3, 4, 5, 6, 7, 8),
  81. each period splits into two, so 4 goes to 8,
  82. 8 goes to 16, and so on,
  83. then we have regions of chaos.
  84. Here, this is aperiodic, with gap in the middle.
  85. This is very narrow but this is a period 5 value
  86. we saw before.
  87. More aperiodic regions.... here's a period 3 gap.
  88. 1, 2, 3 ... I think we investigated that,
  89. maybe back in unit 2,
  90. and then finally up at r = 4, we have
  91. orbits that go from zero to one,
  92. and so it would fill this entire interval.
  93. Ok, so this is the bifurcation diagram for the logistic equation.
  94. We'll spend lots more time exploring this,
  95. but first I would recommend doing the quiz.
  96. It should be quick, and it will just
  97. check your understanding of this lecture,
  98. and then we will look at an online program
  99. that will let you do much, much more exploring
  100. with the bifurcation diagram for the logistic equation.