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Showing Revision 6 created 10/11/2018 by Cigdem Yalcin.

  1. In this section I’ll introduce the idea that a chaotic dynamical system
  2. like the logistic equation what r equals 4
  3. is a deterministic sources of randomness.
  4. In order to do so we’ll have to think carefully about what randomness means.
  5. What does it mean when we say a process or an outcome is random?
  6. I’ll build up a series of arguments layer by layer.
  7. None of these arguments are particularly technical
  8. in the sense of they don’t require calculate or algebra.
  9. However they are conceptually rich and a bit abstract.
  10. But I think we’ll I end up with some really interesting conclusions
  11. that will be perhaps surprising and I hope a lot of fun think about.
  12. So let’s get start it.
  13. I’ll start by introducing a technique known as symbolic dynamics.
  14. The idea behind symbolic dynamics is to convert an orbit, a series of numbers,
  15. in this case between 0 and 1, into a sequence of symbols.
  16. and the standard way to do this is as follows,
  17. if our iterate x is less than 0.5 , I’ll call that L
  18. and if x is greater than or maybe equal to 0.5, I’ll call that R
  19. so I am picturing this would be on the left half of the unit interval
  20. and this is on the right half.
  21. The symbols you use are completely arbitrary
  22. You can use hearts and spades or x and y or zeros and ones.
  23. But I’ll use L and R.
  24. So for example, suppose we had the following itinerary
  25. Ok, so here are the first couple iterates for the logistic equation
  26. Again r equals 4, and initial condition is 0.613
  27. So let’s convert this into symbolic dynamics
  28. So 0.613 that’s greater than a half that would be an R
  29. 0.949 that’s also greater than a half that would be R
  30. 0.194 is less than a half, less than .5, I call then L
  31. 0.625 that’s greater than a half, that’s an R
  32. This is also greater than a half, so that would be an R
  33. So the idea is that I can take any itinerary,
  34. any orbit a sequence of number between 0 and 1
  35. and convert that into a sequence of symbols R R L R R in this case.
  36. So once we have symbols sequence, the idea is that we can study
  37. the dynamics of symbol sequence
  38. instead of the dynamics of the original orbit.
  39. And in many cases, one can show that properties of the orbit
  40. are the same as the properties of the symbol sequence.
  41. So studying the symbol sequence is just as good as the original orbit.
  42. So let me write that
  43. So properties are the same for the orbit and the symbol sequence.
  44. So when I say properties what I mean is
  45. is that say the existence of fixed points and
  46. the stability of fixed points
  47. the symbolical dynamical system involving
  48. just the symbols L and R would have
  49. the same number of fixed points
  50. and their stability would be the same
  51. and if the symbol sequence, the symbolic dynamical system
  52. has say sensitive dependence on initial conditions or aperiodicity
  53. then the original orbit, the original dynamical system would as well.
  54. Now this isn’t an obvious statement at all.
  55. Because it seems like by going to symbols
  56. I am throwing out a lot of information.
  57. After all any number that was between 0 and a half,
  58. I decided to just turn into L
  59. So that’s a very coarse thing to do.
  60. There are lots of lots of numbers and
  61. infinite number of numbers between 0 and a half.
  62. And I just turned all of those into L
  63. So it seems as if I am losing information
  64. so how can these two things be the same.
  65. Well, it turns out that for this particular way of forming symbols
  66. one can show and argue the following
  67. so let me do this with an example of sorts
  68. suppose I show you a symbol sequence R R L R L L R
  69. so then I might ask you what initial conditions could
  70. have given rise to this particular symbol sequence.
  71. And one can show you can kind of infer backwards
  72. to that would correspond to pretty narrow region of initial conditions
  73. and moreover that would just be a single connected region,
  74. that would give rise to this
  75. Then I could say, well, ok, what if the sequence was this
  76. and then you could show that the possible initial conditions
  77. that would have given rise to an orbit
  78. whose symbol sequence is this would be smaller still.
  79. and if I had another symbol, the possible initial conditions
  80. that give rise to this is smaller still
  81. And so in the limit that the symbol sequence becomes infinitely long.
  82. The possible initial conditions that would give rise to it become infinitely small.
  83. Another way to say this is that if you give me one single initial condition
  84. the symbol sequence that results from that is unique.
  85. there is one and only one symbol sequence that
  86. that one that results from that one single initial condition
  87. and that sort of make sense this is a deterministic dynamical system.
  88. So the key feature here is that
  89. there is one to one relationship between initial conditions and symbol sequences.
  90. So if you tell me the infinitely long symbol sequences
  91. I could know, I would know the initial condition
  92. And if I know the initial condition of the deterministic dynamical system
  93. that contains all the information about the orbit.
  94. So the infinite sequence encodes for the initial condition
  95. and the initial condition together with the dynamic tells you the orbit
  96. and from that one can get the properties.
  97. So I guess what I am trying to say is that the information in the symbol sequences
  98. is the same as information in the initial conditions.
  99. And ways of forming symbols from numbers
  100. that have this property are called
  101. generating in the particular scheme is sometimes called generating partition.
  102. So I don’t want to write down a formal definition of this.
  103. Because I think it will get us too far a field
  104. and get us into some really difficult notation
  105. but a partition and a partition was just to go back here
  106. this in a sense would be that a partition the description of the symbolic dynamics
  107. this tells me how to go from the orbit the x’s to the symbols the L and R
  108. that this scheme would be called the generating partition
  109. if longer and longer sequences encode for smaller and smaller
  110. and unique non-overlapping regions of initial conditions.
  111. Ok, so not all symbolic encoding schemes have this nice property
  112. So in fact if I had chose .4 as the cutoff
  113. so if x was less than .4 I call it L
  114. and it’s R otherwise
  115. then that would not have this property
  116. so it’s only special partition special ways of encoding that have this nice feature
  117. but the one that I described does indeed have this nice feature
  118. so it’s only one this is the case that this is true
  119. so let me just to make things little more accurate
  120. say, you know, if we use a generating partition
  121. so provided that we have a generating partition which we do in this case
  122. the properties of the orbit and the properties of symbol sequences
  123. are in the sense I’ve described the same.
  124. Lastly I want to mention that this technique of symbolic dynamics
  125. is a way of proving things about dynamical systems
  126. so I said in the last set of the lectures that it’s proved rigorously that
  127. when r equals 4 the logistic equation has sensitive dependence
  128. on initial conditions and the orbits are aperiodic.
  129. The way one would go about doing that proof
  130. and this is just a very very rough sketch
  131. would be to do this mapping from the dynamical, original dynamical system
  132. to symbol sequences proved properties of these symbol sequences
  133. and then if all of this holds which it would
  134. in this case what you prove about the symbol sequences
  135. which is which are easier to work with
  136. turn out to be true about the orbit as well.
  137. In any event now that we have this idea symbolic dynamics
  138. Let’s take a look at what symbolic dynamics
  139. look like for the logistic equation with r equals 4.