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← Nonlinear 1.2 Quiz solution video

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Showing Revision 2 created 04/13/2021 by jnelson42.

  1. The first problem on this quiz is an
    exercise in evaluating the logistic map,
  2. which looks like this.
  3. Going back to the problem statement,
    we see we're supposed to use
  4. r = 2.5 and x_0 = 0.5
  5. By my calculator, I get 0.625 for that.
  6. To get x_2, we plug that back in, like so.
  7. When I plug that into my calculator, I
    get approximately 0.586.
  8. By the way, what I'm really getting,
    because my calculator has seven
  9. significant figures to the right of the
    decimal place is this
  10. And I'm rounding to
    three significant figures.
  11. To get the third iterate, we plug that
    back in; and when I do that I get
  12. So the answer I get is x_3 = 0.606.
  13. Now this gets pretty painful, so you can
    imagine why it might be nice to have
  14. a computer program to do this.
  15. Here's a simple Matlab
    program that does that.
  16. I'll post this on the supplementary
    materials as well.
  17. As is requested in the quiz problem,
    it takes three inputs:
  18. the first initial condition x_0,
    the parameter value r,
  19. and the number of iterates n.
  20. And it spits out n iterates of the
    logistic map from that x_0 using that r.
  21. So our task is to compute
    the tenth iterate,
  22. starting from x = 0.2, with r = 2.6.
  23. And I'm going to use my Matlab
    program to do that.
  24. And here's the tenth iterate,
    right there: 0.6157.
  25. Here's the answer.
  26. For the third problem, you'll need
    to go to the app that we've written
  27. for you to use to
    explore the logistic map.
  28. You can find that app under the
    supplementary materials section
  29. of the Complexity Explorer
    homepage for this course.
  30. Scroll down to the current unit,
    which is unit 1, and the current segment,
  31. which is segment 1.2, and there's the app.
  32. The task was to plot 50 iterates of the
    logistic map with r = 2
  33. from an initial condition x_0 = 0.2.
  34. Remember, you have to hit the Restart
    Simulation button to get that to work.
  35. The question was, "Does the orbit reach a
    fixed point?"
  36. and that sure looks like it
    reaches a fixed point.
  37. So the answer is definitely yes.
  38. In question four, we're supposed to raise
    r from 2 (where it was in question 3),
  39. up to 2.7 and repeat the same
    experiment: 50 iterates from x_0 = 0.2.
  40. Let's try that.
  41. The shape of the graph is a little bit
    different, but the orbits do
  42. converge to a fixed point.
  43. So the answer is again yes.
  44. The last question asks us, "If the orbits
    in questions 3 and 4 both reached a fixed
  45. point, (which is true), is that fixed
    point at the same value of x?"
  46. So let's look back at the app.
  47. Here's r = 2.7; it looks like the fixed
    point is above this 0.6 line on the plot.
  48. There's r = 2.0; again, still a fixed
    point, but the fixed point is at x = 0.5.
  49. So the answer is no, because those fixed
    points were at different values of x.