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← Chaos 10.1 Summary and Overview (2)

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Showing Revision 1 created 07/17/2016 by sbenham.

  1. Continuing with our review,
  2. the next topics that we covered concern
    bifurcation diagrams.
  3. They're a way to see how the behavior of
    a dynamical system changes as a parameter
  4. is changed.
  5. I think it's best to think of them as
    being built up one parameter value
  6. at a time.
  7. So, for each parameter value, make a phase
    line if it's a differential equation,
  8. or a final-state diagram for an
    iterated function.
  9. And you get a collection of these, and
    then you glue these together to make a
  10. bifurcation diagram.
  11. So, here's one of the first bifurcation
    diagrams we looked at.
  12. This is the logistic equation with
    harvest. The equation is down here.
  13. And so, h is the parameter that
    I'm changing.
  14. H is here, it goes from 0 to 100 to 200
    and so on.
  15. And so, a way to interpret this is
    suppose you want to know what's going on
  16. at h is 100. Well, I would try to focus
    right on that value, and I can see "aha,"
  17. it looks to me like there is an attracting
    fixed point here, and a repelling
  18. fixed point there. So there are two fixed
    points: one of them attracting and one of
  19. them repelling, or repulsive.
  20. And, what's interesting about this is that
    so this is is the stable fixed point,
  21. this would be the stable population of
    the story I told involved fish in a lake
  22. or an ocean, and h is the fishing rate,
    how many fish you catch every year.
  23. And that increases, and as you increase h,
    the sort of steady state population of the
  24. fish decreases, that makes sense.
  25. But what's surprising is that when
    you're here, and you make a tiny increase
  26. in the fishing rate, the steady-state
    population crashes and in fact disappears.
  27. The population crashes.
  28. So, you have a small change in h leading
    to a very large qualitative change in the
  29. fish behavior.
  30. So, this is an example of a bifurcation
    that occurs right here.
  31. It's a sudden qualitative change in the
    system's behavior as a parameter
  32. is varied slowly and continuously.
  33. So, we looked at bifurcation diagrams for
    differential equations and we saw the
  34. surprising discontinuous behavior, then we
    looked at bifurcation diagrams for the
  35. logistic equation, and we saw bifurcations
    here is period 2 to period 4, but what was
  36. really interesting about this was that
    there's this incredible structure to this
  37. and we zoomed in and it looked really
    cool.
  38. There are period 3 windows, all sorts of
    complicated behavior in here.
  39. So, there are many values for which the
    system is chaotic.
  40. The system goes from different period
    to period in a certain way.
  41. And this has a self-similar structure:
    it's very complicated but there is some
  42. regularity to this set of behavior for the
    logistic equation.
  43. So, then we looked at the period doubling
    route to chaos a little bit more closely.
  44. And in particular I defined this ratio,
    delta.
  45. It tells us how many times larger branch n
    is than branch n+1.
  46. So, delta is how much larger or longer
    this is than that.
  47. That would be delta 1. How much longer,
    how many times longer is this length
  48. than that?
    That would be delta 2.
  49. And we looked at the bifurcation diagrams
    for some different functions,
  50. and I didn't prove it, but we discussed
    how this quantity, delta, this ratio
  51. of these lengths in the bifurcation
    diagram is universal.
  52. And that means it has the same value for
    all functions
  53. provided, a little bit of fine print,
    they map an interval to itself and have a
  54. single quadratic maximum.
  55. So, this value, which is I believe
    known to be a rational and I think
  56. transcendental, is known as Feigenbaum's
    constant, after one of the people who made
  57. this discovery of universality.
  58. This is an amazing mathematical fact
    and points to some similarities among
  59. a broad class of mathematical systems.
  60. To me, what's even more amazing is that
    this has physical consequences.
  61. Physical systems show the same
    universality.
  62. So, the period doubling route to chaos
    is observed in physical systems.
  63. I talked about a dripping faucet and
    convection rolls in fluid, and one can
  64. measure delta for these systems.
    It's not an easy experiment to do,
  65. but it can be done.
  66. And the results are consistent with this
    universal value, 4.669.
  67. And so, what this tells us is that somehow
    these simple one-dimensional equations,
  68. we started with a logistic equation,
    an obviously made-up story about
  69. rabbits on an island, that nevertheless
    produces a number, a prediction
  70. that you can go out in the real physical
    world and conduct an experiment
  71. with something much more complicated
    and get that same number.
  72. So, this is I think one of the most
    surprising and interesting results
  73. in dynamical systems.
  74. So, then we moved from one-dimensional
    differential equations to two-dimensional
  75. differential equations.
  76. So now, rather than just keeping track
    of temperature or population, we're
  77. going to keep track of two populations,
    say R for rabbits and F for foxes.
  78. And we would have now a system of two
    coupled differential equations:
  79. the fate of the rabbits depends on
    rabbits and foxes, and the fate of the
  80. foxes depends on foxes and rabbits.
  81. So, they're coupled, they're
    linked together.
  82. And one can solve these using Euler's
    method or things like it,
  83. very, almost identically to how one would
    for one-dimensional differential equations
  84. And you get two solutions:
  85. you get a rabbit solution and a fox
    solution.
  86. And in this case, this is the
    Lotka-Volterra equation,
  87. they both oscillate.
    We have cycles in both rabbit and foxes.
  88. But then, we could plot R against F.
  89. So, we lose time information, but it will
    show us how the rabbits and the foxes
  90. are related.
  91. And if we do that, we get a picture that
    looks like this.
  92. Just a reminder that this curve goes in
    this direction.
  93. And so, the foxes and rabbits are
    cycling around.
  94. The rabbit population increases,
    then the fox population increases.
  95. Rabbits decrease because the foxes
    are eating them.
  96. Then the foxes decrease because
    they're sad and hungry because
  97. there aren't rabbits around, and so on.
  98. So, this is is similar to the phase line
    for one-dimensional equations,
  99. but it's called a phase plane because it
    lives on a plane.
  100. And this hows how R and F are related.
  101. Phase plane and then phase space is
    one of the key geometric constructions,
  102. analytical tools used to visualize
    behavior of dynamical systems.
  103. So, an important result is that there
    can be no chaos, no aperiodic
  104. solutions in 2D differential equations.
  105. So, curves cannot cross in phase space.
  106. The equations are deterministic, and that
    means that every point in space, and
  107. remember this is in phase space, so my
    point in space gives the rabbit and fox
  108. population, there's a unique direction
    associated with the motion.
  109. DF/DT, DR/DT, that gives a direction.
    It tells you how the rabbits are
  110. increasing, how the foxes are increasing.
  111. If two phase lines ever cross, like they
    do where my knuckles are meeting,
  112. then that would be a non-deterministic
    dynamical system.
  113. There would be two possible trajectories
    coming from one point.
  114. So, the fact that two curves can't cross
    in these systems limits the behavior.
  115. They sort of literally paint themselves in
    as they're tracing something out, tracing
  116. a curve out in phase space.
  117. So there can be stable and unstable fixed
    points and orbits can tend toward infinity
  118. of course, and there can also be limit
    cycles attracting cyclic behavior, and we
  119. saw an example of that.
  120. But the main thing is that there can't be
    aperiodic orbits.
  121. And that result is known as the
    Poincaré-Bendixson theorem.
  122. It's about a century old.
  123. And it's not immediately obvious;
    it takes some proof.
  124. Like I said, that's maybe why it's a
    theorem and not just an obvious statement.
  125. One could imagine, and people in the
    forums have been trying to imagine
  126. space-filling curves that somehow never
    repeat but also never leave a bounded area
  127. But the Poincaré-Bendixson theorem says
    that those solutions somehow
  128. aren't possible.
  129. So, the main result is that
    two-dimensional differential equations
  130. cannot be chaotic.
  131. That's not the case for three-dimensional
    differential equations, however.
  132. So, here are the Lorenz equations.
  133. Now, again it's a dynamical system, it's
    a rule that tells how something changes
  134. in time.
  135. Here that something is x, y, and z, and
    I forget what parameter values I chose
  136. for sigma, rho, and beta.
  137. And we can get three solutions:
    x, y, and z.
  138. And these are all curves plotted as a
    function of time.
  139. But we could plot these in phase space,
    x, y, and z together.
  140. And for that system, if we do that, we get
    some complicated structure that
  141. loops around itself and repeats.
  142. It looks like the lines cross, but
    they don't.
  143. There's actually a space between them.
  144. It looks like they cross because this is
    a two-dimensional surface trying to
  145. plot something in 3D.
  146. Alright, so just a little bit more about
    phase space.
  147. Determinism means that curves in phase
    space cannot intersect.
  148. But because the space is three-dimensional
    curves can go over or under each other.
  149. And that means that there is a lot more
    interesting behavior that's possible.
  150. A trajectory can weave around and under
    and through itself
  151. in some very complicated ways.
  152. What that means, in turn, is that
    three-dimensional differential equations
  153. can be chaotic.
  154. You can get bounded, aperiodic orbits,
    and it has sensitive dependence as well.
  155. And then we saw that chaotic trajectories
    in phase space are particularly
  156. interesting and fun.
  157. They often get pulled to these things
    called strange attractors.
  158. So here's the Lorenz attractor or the
    famous values for the Lorenz equation.
  159. Strange attractors.
    What are strange attractors?
  160. Well, they're attractors, and what that
    means is that nearby orbits get
  161. pulled into it.
  162. So, if I have a lot of initial conditions,
    they all are going to get
  163. pulled onto that attractor.
  164. So, in that sense, it's stable.
    If you're on that attractor and somebody
  165. bumps you off a little bit, you'd get
    pulled right back towards it.
  166. That's what it means to be stable.
  167. So, it's a stable structure
    in phase space.
  168. But the motion on the attractor is not
    periodic the way most attractors that
  169. we've seen are, or even fixed points.
  170. But the motion on the attractor
    is chaotic.
  171. So, once you're on the attractor, orbits
    are aperiodic and have
  172. sensitive dependence on
    initial conditions.
  173. So, it's an attracting chaotic attractor.
  174. Then we looked at this a little bit more
    geometrically and I argued that the key
  175. ingredients to make a strange attractor
    or to make chaos of any sort, actually,
  176. is stretching and folding.
  177. So, you need some stretching to pull
    nearby orbits apart.
  178. The analogy I discussed was
    kneading dough.
  179. So, when you knead dough, you stretch it.
  180. That pulls things apart, and then you fold
    it back on itself.
  181. So, the folding keeps orbits bounded.
  182. It takes far apart orbits and moves them
    closer together.
  183. But stretching pulls nearby orbits apart,
    and that's what leads to
  184. the butterfly effect,
    or sensitive dependence.
  185. Now, stretching and folding, it may be
    relatively easy to picture in
  186. three-dimensional space, either a space
    of actual dough on a bread board
  187. or a phase space.
  188. But it occurs in one-dimensional maps
    as well, the logistic equation stretches
  189. and folds.
  190. And this can explain how one-dimensional
    maps, iterated functions, can capture
  191. some of the features of these
    higher dimensional systems.
  192. And it begins to explain, also, how these
    higher dimensional systems,
  193. convection rolls, dripping faucets,
    can be captured by one-dimensional
  194. functions like the logistic equation and
    this universal parameter, 4.669.
  195. So, in any event, stretching and folding
    are the key ingredients for a chaotic
  196. dynamical system.
  197. So, strange attractors once more.
    They're these complex structures that
  198. arise from simple dynamical systems.
  199. A reminder that we looked at three
    examples: the Hénon map, the Hénon
  200. attractor, which is a two-dimensional
    discrete, iterated function.
  201. And then, two different sets of coupled
    differential equations in three dimensions
  202. the famous Lorenz equations, and also
    the slightly less famous but equally
  203. beautiful Rössler equations.
  204. Again, the motion on the attractor is
    chaotic, but all orbits get pulled
  205. to the attractor.
  206. So, strange attractors combine elements
    of order and disorder.
  207. That's one of the key themes
    of the course.
  208. The motion on the attractor is locally
    unstable.
  209. Nearby orbits are getting pulled apart,
    but globally it's stable.
  210. One has these stable structures, the same
    Lorenz attractor appears all the time.
  211. If you're on the attractor, you get pushed
    off it, you get pulled right back in.
  212. Alright, and the last topic we covered in
    unit 9 was pattern formation.
  213. So, we've seen throughout the course in
    the first 8 units that dynamical systems
  214. are capable of chaos.
  215. That was one of the main results.
  216. Unpredictable, aperiodic behavior.
  217. But there's a lot more to dynamical
    systems than chaos.
  218. They can produce patterns, structure,
    organization, complexity, and so on.
  219. And we looked at just one example
    of a pattern-forming system.
  220. There are many, many ones to choose from.
  221. But we looked at reaction-diffusion
    systems.
  222. So there, we have two chemicals that
    react and diffuse.
  223. And diffusion, that's just the random
    spreading out of molecules in space,
  224. diffusion tends to smooth out
    differences, it makes everything
  225. as boring and bland as possible.
  226. But, if we have two different chemicals
    that react in a certain way,
  227. it's possible to get stable spatial
    structures even in the presence
  228. of diffusion.
  229. Here are these equations-- I described
    them in the last unit.
  230. This is deterministic, just like the
    dynamical systems we've studied before.
  231. And it's spatially extended, because now
    U and V are functions, not just of T, but
  232. of X and Y.
  233. So, these become partial differential
    equations.
  234. Crucially, the rule is local.
  235. So, the value of U or the value of V,
    those are chemical concentrations,
  236. depend on some function of a current
    value at that location, and on this
  237. Laplacian derivative at that location.
  238. So, we have a local rule in that the
    chemical concentration here doesn't know
  239. directly what the chemical concentration
    is here; it's just doing it's own thing
  240. at its own local location.
  241. Nevertheless, it produces these
    large-scale structures.
  242. So, just one quick example.
  243. We experimented with reaction-diffusion
    equations at the
  244. Experimentarium Digitale site.
  245. Here's an example that we saw emerging
    from random initial conditions,
  246. these stable spots appear.
  247. And then we also looked at a video from
    Stephen Morris at Toronto where two fluids
  248. are poured into this petri dish, and like
    magic, these patterns start to emerge
  249. out of them.
  250. So, Belousov Zhabotinsky has another
    example of a reaction-diffusion system.
  251. So, pattern formation is a giant subject.
  252. It could be probably a course in and of
    itself.
  253. The main point I want to make is that
    there's more to dynamical systems
  254. than just chaos or unpredictability
    or irregularity.
  255. Simple, spatially-extended dynamical
    systems with local rules are capable
  256. of producing stable, global patterns and
    structures.
  257. So, there's a lot more to the study of
    chaos than chaos.
  258. Simple dynamical systems can produce
    complexity and all sorts of interesting
  259. emergent structures and phenomena.