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← Chaos 5.2 The Bifurcation Diagram (1)

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Showing Revision 24 created 10/29/2018 by Cigdem Yalcin.

  1. So we now have a collection of final state diagrams.
  2. They’re the examples that I did in the last video
  3. 3.2 , 2.9 and then an aperiodic value at 3.8
  4. and then for the quiz that perhaps you just took
  5. you did 3.4 that’s another period 2 value
  6. 3.739 this is period 5
  7. 1,2,3,4,5 1,2,3,4,5
  8. and then another aperiodic one 3.9
  9. for here the dots extend from about .1 to almost 1 exactly, may be .98
  10. so it might look something like this
  11. so we got a collection of final state diagrams
  12. and as we did with differential equation
  13. we’ll form a bifurcation diagram
  14. by gluing together a collection of final state diagrams
  15. so let me take each of these and cut it out
  16. so I can move it around so how do you that and we’ll see what it looks like
  17. so now I’ve got a collection of final state diagrams
  18. and let’s put them in order and as before put them on the side as they do that
  19. so there is 2.9 that was period 1
  20. here’s 2.3 that’s a stable cycle of period 2
  21. here is 3.4
  22. 3.739
  23. 3.8
  24. and 3.9
  25. so this is the beginning of a bifurcation diagram for the logistic equation
  26. remember the goal of the bifurcation diagram is to see
  27. how the dynamical systems behavior changes
  28. as a parameter in this case r is changed
  29. so it gives us a global view of the range of behaviors
  30. that dynamical systems can exhibit
  31. in this case with just these six final state diagrams
  32. it’s not really clear quite yet what the overall pattern might be
  33. so in order to sort of see a pattern to connect the dots so to speak
  34. we would need to try this out from many many more r values
  35. and make many more phase lines and stack them all in here densely
  36. so that we can see what happens from one r value to the next
  37. so as you’ve probably guessed I’ll use a computer to do that work for us
  38. and I’ll show you the program and how it works in a little bit
  39. but first let’s focus on what the results are
  40. so let’s see put these to the side just for a moment
  41. here’s the bifurcation diagram for the logistic equation
  42. the lower limit here is r equals 0
  43. and here’s r equals 4
  44. and then this goes from 0 to 1
  45. so when r is between 0 and 1
  46. between my fingers here, the, can’t quite see it here but the only fixed point is
  47. the attracting fixed point is 0
  48. if the growth rate is less than 1 the rabbits die out
  49. between 1 and 3 there is an attracting fixed point
  50. and in fact we saw that let’s see we did one for 2.9 here it is
  51. let’s see if this is going to work,
  52. close, ok so 2.9 that’s about there and this dot that I drew in the first example
  53. is part of this line here then as r is increased
  54. as the growth rate gets larger and larger
  55. the period 1 behavior splits into period 2
  56. and we’ve seen that here’s r equals 3.2
  57. r equals 3.2 so the two dots from the final state diagram
  58. show up as part of this line here
  59. so in this region where if I go up from a single point I see two lines two dark regions
  60. that would indicate that its period 2
  61. here is another r value a little bit larger 3.4
  62. and still period 2, there are only 2 dots
  63. but the periods are little bit further apart
  64. see if I can get both of these on at the same time
  65. so for these two different values, it’s the same qualitative behavior
  66. attracting cycle of period 2 but the exact locations are a little bit different
  67. all right it’s little hard to see what’s going on in here
  68. so we’ll zoom in here in just a moment
  69. but first just a little bit of terminology which should be familiar
  70. from what we did with differential equations
  71. I’d say the system undergoes a bifurcation here r equals 3
  72. remember a bifurcation is a sudden qualitative change in the behavior of a dynamical system
  73. as a parameter is varied continuously
  74. so the qualitative change here is that the fixed point here splits into two
  75. so we go from an attractor of period 1 to an attractor of period 2
  76. so that’s a bifurcation and it’s called a period doubling bifurcation
  77. because the period doubles
  78. here we see we have a bifurcation from period 2 to period 4
  79. so that’s another period doubling bifurcation
  80. ok, let’s zoom in on the bifurcation diagram,
  81. let’s look at just to this portion, let’s look at what’s going on from 3 to 4
  82. since this is where a lot of the interesting action is
  83. so here I’ve zoomed in and this is a bifurcation diagram from 3 to 4
  84. so we see in this region from 3 to about a little more than 3.4
  85. the behavior is period 2, here are the two phase lines we final state diagrams we drew
  86. previously there’s 3.2 and there’s 3.4 and they line up pretty well
  87. let’s see if I can get a few more on here
  88. here’s 3.739 and that corresponds to this funny region here
  89. this light region we’ll look at that more closely in a bit
  90. but period 5 1,2,3,4,5
  91. and then we had 2 aperiodic values at 3.8
  92. and around there that looks pretty good
  93. and then 3.9 which is right around there
  94. so the bifurcation diagram for the logistic map looks quite different
  95. then the ones we saw for differential equations
  96. which isn’t surprising the logistic the logistic map and things like it
  97. exhibit chaos aperiodic behavior
  98. so we’ll expect it to be more richer bifurcation diagram
  99. and have more features to look at
  100. but remember the thing about bifurcation diagrams to interpret them
  101. remember that they began their life as a series of in this case final state diagrams
  102. so for example if I wanted to know what’s going on right around 3.7
  103. I would just try to blot out everything except for 3.7
  104. and then view it as a single final state diagram
  105. sort of imagine doing that with this, thing that I’ve made
  106. so this is I’ve moved this so that the split shows right around 3.7
  107. and so we would say that ahaa this looks like an aperiodic region
  108. lots and lots of dots so it must be aperiodic
  109. going from between this value and this value
  110. if I want to know what’s going on at 3.2
  111. I could move this until I’m seeing 3.2
  112. and then I would see just these 2 dots here or small line segments
  113. and that would mean that this is periodic with period 2
  114. you can imagine another way to view this as r increases
  115. you see period 2 behavior and the two values are getting further apart
  116. they’re moving this way as I let r get larger
  117. and then a little passed 3.4
  118. that’s where is it there it is, there’s a bifurcation,
  119. so now it’s period 4 1,2,3,4
  120. like small change I go then a small change in r that’s moving this
  121. leads to a qualitative change in the behavior of the dynamical system
  122. in this case we go from 2, a cycle of period 2 to a cycle of period 4
  123. and then as I increase r further still there’s a region of period 8
  124. 1,2,3,4,5,6,7,8 each period splits into 2 so 4 goes to 8, 8 goes to 16 and so on
  125. then we have regions of chaos here this is aperiodic but with a gap in the middle
  126. this is it’s very narrow but this is the period 5 value we saw before
  127. more aperiodic regions, here’s a period 3 gap 1,2,3
  128. I think we investigated that maybe back in Unit-2
  129. and then finally up at r equals 4 we have orbits that go from 0 to 1
  130. so it would fill this entire interval
  131. ok, so this is the bifurcation diagram for the logistic equation
  132. we’ll spend lots more time exploring this
  133. but first I would recommend doing the quiz it should be quick
  134. and it will just kind of check your understanding of this lecture
  135. and then we will look at an online program
  136. that will let you do much much more exploring
  137. with the bifurcation diagram for the logistic equation