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← 04 05HysteresisHD

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Showing Revision 3 created 07/06/2015 by Gary Hlusko.

  1. In this optional sub unit
  2. I'll present the bifurcation diagram
  3. for a different differential equations
  4. and this will lead us to
  5. the phenomenon of Hysteresis or path dependence
  6. we will see that in a second.
  7. We will start with this differential equation
  8. dx/dt. I'll use x this time
  9. instead of P.
  10. because this doesn't really represent a population
  11. is rx plus x cubed minus x to the fifth
  12. So r is now our parameter.
  13. Before it was h.
  14. This time we will use r.
  15. So, we will build up the bifurcation diagram
  16. piece by piece
  17. by letting r be different values
  18. plotting the right hand side of this
  19. and seeing what the function looks like
  20. and making a phase line
  21. So here is what we have if r equals one
  22. Down, up, and down
  23. So there are three fix points.
  24. Because the line crosses the x axis three times
  25. here, here, and here.
  26. So, three fixed points.
  27. One, two, three.
  28. make a note this is for r equals one
  29. When this function is negative.
  30. This is a derivative.
  31. The derivative is negative.
  32. X is decreasing
  33. When positive we are increasing.
  34. Negative decreasing, positive increasing
  35. So, this function has three fixed points.
  36. There is an unstable fixed point at zero.
  37. and there are two stable fixed points
  38. out here a little bit more than
  39. one away from the origin.
  40. So that's the situation when R equals 1.
  41. If I decrease R and make it a little
  42. bit negative.
  43. This curve gets a little wiggle in it.
  44. and it starts to look like this
  45. So the curve gets steeper
  46. but it aquires a little wiggle in here.
  47. So let's calculate
  48. let's figure out the phase line for this
  49. here we have five fixed points.
  50. 1,2,3,4,5
  51. equilibria class of five.
  52. and they kind of scrunch together
  53. That's going to be a little challenging
  54. for me to draw.
  55. Ok, so there are the fixed points.
  56. 1,2,3,4,5.
  57. the function is positive
  58. so we are moving to the right
  59. negative in here, then positive
  60. negative,positive, negative.
  61. R equals zero point two.
  62. So I see three stable fixed points.
  63. Here, here, and here in the middle
  64. So you have probably noticed
  65. a stable fixed point occurs
  66. when the line crosses the axis from top to bottom.
  67. So that happens here, here, and here.
  68. So we have these two unstable fixed points
  69. here and here.
  70. When the line goes from below to above.
  71. So, five fixed points, three are stable
  72. and two are unstable.
  73. This is the story for minus 0.2
  74. the last r we will look at
  75. is r equals minus 0.4
  76. R is a little bit more negative here.
  77. and what happens is
  78. these bumps straighten out.
  79. So this bump and this bump
  80. get pulled up and down.
  81. And we end up with this.
  82. So here, the phase line is kind of simple
  83. almost boring again
  84. So we have one fixed point.
  85. So we had five but four of them disappeared.
  86. And we are just left with this one.
  87. at the origin.
  88. And it keeps --it's stability
  89. so we had a little hard to see.
  90. We had four and here we had one.
  91. but this one the one at the origin remains.
  92. Ok, so we have three phase lines.
  93. So we can connect them
  94. Sort of glue them all together
  95. and see what the bifurcation diagram might look like.
  96. so as before I"m going to slice off.
  97. these phase lines.
  98. and let's take a look.
  99. Here is R equals 1
  100. Here is R equals minus 0.2
  101. And I should've written here
  102. this was r equals minus 0.4
  103. Here is the what we have.
  104. So from these phase lines
  105. it might not be immediately clear
  106. what the entire bifurcation diagram looks like
  107. we might want to do a few more phase lines
  108. For immediate R values.
  109. try an R of 0. a R of -.1
  110. A r of +.1, and so on
  111. But rather than take the time to do that.
  112. Let me sketch what this looks like
  113. and then I'll show you a neater drawing
  114. of the bifurcation
  115. diagram
  116. Since the main goal is to get this bifurcation diagram
  117. and then look at it and learn about Hysteresis
  118. so let me just draw a few things on here.
  119. So I'm going to use blue
  120. for an unstable fixed point
  121. and so it turns out I have a line
  122. of unstable fixed points here.
  123. Wait sorry those are stable.
  124. Oh, dear how can I recover from this
  125. this was going to be blue
  126. Maybe it's red and blue, purple, or it looks mostly red
  127. So these are stable.
  128. It's just the wrong color
  129. It's stable the arrows are going in
  130. and then we also have some stable fixed points
  131. Here and here. Here and Here.
  132. and these are going to look like this.
  133. And this one is going to come down like this.
  134. and then we will have unstable fixed point here.
  135. and this line connects up here.
  136. So that's our bifurcation diagram
  137. It's not the best picture in the world.
  138. To me, it kind of looks like
  139. a fish like a salmon that's throwing up.
  140. Which you know.
  141. is not what I intended.
  142. but this is the bifracation diagram.
  143. So we have stable points in red
  144. and unstable points in blue.
  145. And hopefully you can see how the blue and red lines
  146. line up with these fixed points.
  147. And this vommiting fish looking this.
  148. So let me draw another nicer version of this diagram
  149. and we will analyze that.
  150. And learn about Hysteresis.
  151. So here is a slightly neater version
  152. of the bifucation diagram.
  153. From the previous screen.
  154. And I'll be focusing on the positive x-values
  155. I've only drawn arrows on here.
  156. So we have a line of stable fixed points.
  157. Attractors.
  158. and we have here in blue a line of unstable
  159. fixed points repellers.
  160. Unstable here, and stable here.
  161. So, let's imagine let's sort of talk through
  162. a scenario with this.
  163. That the parameter starts off somewhere off here.
  164. And we have a postive x value.
  165. We are going to get pulled to this attractor
  166. and now imagine the parameter
  167. is going to be decreasing
  168. who knows in this case.
  169. I don't know if there is a clear physical or analogue
  170. or something but whatever R is. It decreases
  171. So as R decreases then the equilibrium
  172. value decreases.
  173. then we decrease R some more
  174. and the equilibrium value decrease some more
  175. then we move down along here.
  176. And this looks alot like
  177. what happened when we were increasing
  178. the fishing rate in the logistic differential equation
  179. So we move down here,
  180. R continues to decrease
  181. R continues to decrease
  182. R continues to decrease
  183. until we get here.
  184. And then this fixed point
  185. this attractor up here disappears
  186. It's gone.
  187. It decrease a little bit more.
  188. The quantity of x whatever it is
  189. is going to get pulled down here to zero.
  190. And so then,
  191. perhaps we like this positive thing
  192. is good
  193. zero is bad
  194. maybe this is growth rate of the economy
  195. or some fishing, some number of fish
  196. or something
  197. and we zip down here.
  198. Then we might say "Uh-oh, we crashed"
  199. "We better increase R."
  200. and so we will increase R.
  201. but this red point down here is stable.
  202. it's attracting
  203. And so we don't automatically jump up to here.
  204. because this is stable.
  205. We move a little bit
  206. We get pushed back.
  207. So then we would increase R,
  208. We will increase R,
  209. we will increase R still.
  210. More, until we get a little bit over here.
  211. Then. this fixed point loses it's stability.
  212. We go from Red to blue.
  213. and then we will jump back up to here.
  214. So again, we are seeing jumps
  215. But this time there is a new feature.
  216. Which is as follows:
  217. Suppose we wanted to know if R was around here.
  218. Whatever that is -0.2
  219. What stable behavior would we observe in this model
  220. and the answer is,
  221. it would depend on not just
  222. on the R value, but where one came from.
  223. and this is the idea of the Hysteresis.
  224. Let me draw a picture sort of to illustrate.
  225. or outline the story I just told.
  226. So thinking of this portion of the bifurcation diagram
  227. I guess I'll just make a really rough sketch of this.
  228. So, I could move down this way
  229. then I come to this collapse point
  230. and I go down here.
  231. Then, I would increase until here.
  232. and then I would jump back up
  233. and could go in either direction here.
  234. So, so this is to connect it r = 0
  235. So this system so has path dependence.
  236. So what would you observe at this r value
  237. Well it depends not just on the R value.
  238. but on the path to get there.
  239. If you reach this R value,
  240. the one where my finger is
  241. from above, from the right
  242. Then you would be up here.
  243. Here on this diagram.
  244. If you approached this R value from below
  245. having going beyond this and sort of falling off that cliff
  246. then you will be down here at zero
  247. this is called hysteresis or path dependence.
  248. So the term of this behavior
  249. is hysteresis or path dependence.
  250. So that the equilibrium property
  251. the oberseved behavior of this differential equation
  252. this model
  253. depends not only on R.
  254. It looks like it only depends on R.
  255. If you tell me what R is.
  256. I can solve the differential equation
  257. I can tell you what X would end up being.
  258. But in the situation where you have multiple attractors
  259. and they are arranged like this
  260. knowing R is not enough
  261. you need to know where R came from.
  262. It depends not just on R.
  263. But on the path R took.
  264. This is surprising and interesting
  265. I think because path dependence
  266. is a type of memory
  267. The value of the population
  268. whatever this is
  269. in a sense remembers where its been.
  270. It's not obvious at all that this equation has memory
  271. built into it
  272. This says the growth rate,
  273. the change of X and this number R.
  274. So it's a type of memory or history
  275. that get introduced into a differential equation
  276. as a result of this bifurcation
  277. this particular structure in a bifurcation diagram
  278. like this.
  279. I don't know that this
  280. is common or ubiquitous in differential equations
  281. But it's not uncommon either
  282. But you don't need a tremendous complicated equation
  283. to get this behavior
  284. So this is another type-I guess--
  285. of bifucations
  286. Two bifucations.
  287. There is a bifurcation here
  288. and a bifurcation there.
  289. and taken together
  290. those two bifurcations lead to this path dependence.
  291. So, again to underscore it one more time
  292. We have a simple differential equation
  293. something that is continuous, smooth, differential,
  294. doesn't have any memory built in
  295. and we can have a system behave in jumps
  296. and that develops a memory or path dependence
  297. So that's the idea behind.
  298. Hysteresis or path dependence