 ## ← Chaos 4.1 The Logistical Diffierential Equation (2)

• 1 Follower
• 166 Lines

### Get Embed Code x Embed video Use the following code to embed this video. See our usage guide for more details on embedding. Paste this in your document somewhere (closest to the closing body tag is preferable): ```<script type="text/javascript" src='https://amara.org/embedder-iframe'></script> ``` Paste this inside your HTML body, where you want to include the widget: ```<div class="amara-embed" data-url="http://www.youtube.com/watch?v=gwPV9vHB4Hc" data-team="complexity-explorer"></div> ``` 3 Languages

• English [en] original
• Arabic [ar]
• Spanish, Argentinian [es-ar]

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.)
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.