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← Chaos 4.1 The Logistical Diffierential Equation (2)

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Showing Revision 8 created 09/25/2014 by JULIE BEAL.

  1. In this video, I'll compare and contrast
  2. differential equations
    and iterated functions.
  3. These are the two main types
    of dynamical systems
  4. that we'll study in this course,
  5. and, although they're very similar,
  6. they do have some different
    mathematical properties,
  7. and comparing the logistic equation,
  8. as an iterated function
    and differential equation
  9. can help make this clear,
  10. and highlight some important distinctions.
  11. So here on the left, is the logistic
    differential equation.
  12. A differential equation
    (the form we'll be studying)
  13. describes a function P
  14. in terms of its rate of change.
  15. So this says, we know
    the rate of change of P
  16. if we know what P is.
  17. The population growth
    depends on the population value
  18. in these two parameters.
  19. For an iterated function,
  20. it also describes a population growth,
  21. but here, f(P) is
    the population next year,
  22. given the population P this year.
  23. So we get a series of population values
  24. by iterating this function.
  25. So I began when I derived
    the logistic equation,
  26. (I used this form),
  27. but it's often simplified to this,
  28. the A kind of gets absorbed inside x.
  29. So this is what we worked with,
  30. but the starting point
    for these two equations
  31. is the same on the right-hand side.
  32. What's different is, we interpret things
  33. differently on the left-hand side.
  34. So the right-hand side here
  35. is interpreted as the growth rate.
  36. The right-hand side here
  37. is interpreted as
    the population next year.
  38. So solutions to these iterated functions
    and differential equations
  39. have a different character.
  40. For differential equations,
  41. the solution is population
    as a function of time,
  42. and that would look, as we've seen,
  43. maybe something like this.
  44. For an iterated function,
  45. we end up with a time series plot,
  46. and that might look
  47. something like this.
  48. So notice the difference
  49. between these two solutions.
  50. In both cases, the blue curve
  51. is the solution to the dynamical system.
  52. The dynamical system is just a rule
  53. that tells this blue thing what to do.
  54. But for the differential equation,
  55. the blue curve changes continuously.
  56. It's defined at all times,
  57. and it smoothly increases,
  58. say, from here to here,
  59. and it has to pass through all
    intermediate values.
  60. For the iterated function,
  61. the time moves in jumps.
  62. It has an initial value at time 0,
  63. then time 1,
  64. then time 2,
  65. and the value of the population
    also moves in jumps.
  66. It goes from this value
  67. to this value,
  68. and even though we connect those dots,
  69. it doesn't slide through
    all values in between.
  70. It jumps from here at time 0
  71. to here at time 1
  72. without going through
    the intermediate values.
  73. In this one, the differential equation,
  74. time and the population are continuous.
  75. Time and population are continuous.
  76. But for the logistic equation
  77. and all iterated functions,
  78. the time and the population
    or whatever we're measuring
  79. moves in jumps.
  80. So, again, for the logistic equation
    and the iterated function,
  81. time and population moves in jumps.
  82. And this difference here,
  83. together with the fact that
    these equations are deterministic,
  84. gives rise to very different ranges
    of possible behaviors.
  85. So we've seen for the iterated function
  86. in Unit 3
  87. that it's capable of producing
    cycles and chaos.
  88. So cycles and chaos are possible.
  89. Of course not all iterated functions
  90. will show a cycle or will show chaos
  91. and remember chaos is
    an aperiodic bounded orbit
  92. that also has sensitive dependence
    on initial conditions.
  93. For a differential equation, however,
  94. cycles and chaos are not possible.
  95. So let's think about why this is so.
  96. So suppose a cycle was possible.
  97. If that was the case,
  98. I would have a solution curve
  99. that looked something like that.
  100. It goes up and down.
  101. We can eliminate this possibility
  102. by appealing to the determinism
    of this equation.
  103. This equation says that the derivative,
  104. the growth rate, the rate of change
    of the population, depends
  105. only on the population.
  106. (And r and K, but we're imagining
    those are fixed.)
  107. So let's think about this blue curve here
  108. that oscillates up and down.
  109. I'm going to draw, just arbitrarily,
  110. a dashed line through here.
  111. And notice what happens.
  112. Here, I have a particular p value,
  113. the p value is at this dashed line,
  114. and the population is increasing
  115. so the derivative is positive.
  116. The derivative is positive
    for this p value.
  117. Over here, when the population
    is going back down,
  118. the population is decreasing,
  119. so the derivative is negative.
  120. So that means at these two points,
  121. here and here,
  122. they're different derivatives.
  123. So at the first purple arrow
  124. the function is increasing :
    positive derivative.
  125. At the second arrow
  126. the function is decreasing:
    negative derivative.
  127. But the problem is that
  128. they have the same p value
    as on the y axis here.
  129. And the p value is the same.
  130. If this was true, this would say
  131. different derivatives at the same p value.
  132. But that's impossible
  133. because the differential equation says
  134. the derivative is a function
    of only the p value.
  135. Another way of saying that is
  136. a given p value only has one
    derivative associated with it.
  137. If you know the population p
  138. then that determines the derivative.
  139. Here, if you know the population p
  140. that does not determine the derivative,
  141. because you have different derivatives
  142. at the same p value.
  143. So the conclusion, then, is
  144. that cycles are not possible,
  145. and chaos isn't possible as well.
  146. Any behavior that goes up and down
  147. (it doesn't have to be a regular cycle)
  148. we can eliminate by this argument.
  149. As we said in Unit 2
  150. the range of behaviors for one dimensional
    differential equations
  151. are kind of boring.
  152. The function can increase to a fixed point
  153. decrease to a fixed point,
  154. decrease to infinity,
    increase to infinity,
  155. and that's all it can do.
  156. Iterated functions have a much richer
    array of behavior,
  157. and that's because determinism
    doesn't constrain them
  158. in the same way, so that
    it doesn't forbid cycles.
  159. So cycles are possible
    in iterated functions
  160. and chaos, aperiodic behavior,
    is possible as well.
  161. In the next sub-unit
  162. we'll leave iterated functions
  163. behind for a little bit
  164. and we'll look again at the logistic
    differential equation
  165. and I'll add a term to it
  166. and we'll start
    investigating bifurcations.