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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.
8. dx/dt. I'll use x this time
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.
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.
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
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
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