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← Nonlinear 1.3 Transients and attractors

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Showing Revision 3 created 07/25/2016 by Steven Gunawan.

  1. In the last segment, you saw that the progression
    of iterates of the logistic map converged to an
  2. asymptote. In this segment, I'm going to be a
    bit more careful about the definitions and
  3. terminology around all of that. And I'm
    going to show you what happens for different
  4. values of the initial conditions x_0 and the
    parameter, R.
  5. First of all, that notion of a progression
    of iterates, x_0, x_1 and so on.
  6. That's called an orbit or a trajectory of
    the dynamical system.
  7. An orbit or a trajectory is a sequence of
    values of the state variables of the system.
  8. The logistic map has one state variable, x. Other
    systems may have more than one state variable
  9. My pendulum, for instance, the one you saw
    in the very first segment. You need to know
  10. the position and velocity of both bobs of
    the pendulum in order to say what state it's in.
  11. I'll come back to that in the third unit of
    this course.
  12. The starting value of the state variable in the
    logistic map, x_0, is called the initial condition.
  13. The trajectory of the logistic map from the
    initial condition, x=0.2 with R=2, reaches
  14. what's called a fixed point. That's the asymptote
    after going through what's called a transient.
  15. I drew that picture for you last time.
    Here's that picture again.
  16. Technically, a fixed point is a state of the
    system that doesn't move under the influence
  17. of the dynamics. That is, the fixed point to
    which the logistic map orbit converges, is
  18. what's called an attracting fixed point.
  19. There are other kinds of fixed points as
    I'll show you with my pendulum.
  20. So this is certainly a fixed point of the
    dynamics. The system is there and the
  21. dynamics are not causing it to move. And it's
    an attracting fixed point, because if I perturb
  22. it a little bit, that perturbation will shrink,
    returning the device to the fixed point.
  23. Now, that's an attracting fixed point. As I
    said, there are other kinds of fixed points.
  24. This is one of them. Or, there is one here.
    I've never gotten the pendulum to sit at it.
  25. There is some point here for the pendulum
    where it will balance.
  26. So that is a fixed point in the sense that
    the system will not move from there,
  27. but it is an unstable fixed point.
  28. There are two other unstable fixed points
    in this system. This one, and this one.
  29. Again, all of these points are states of the
    system that the dynamics is stationary.
  30. This definition that I just gave you captures
    both kinds of fixed points.
  31. States that don't move under the influence
    of the dynamics, but doesn't tell you whether
  32. they are stable, that is, they are attracting,
    or they are unstable, that is, they are repelling,
  33. like the inverted point of the pendulum.
  34. Dynamical systems have several different kinds
    of asymptotic behaviors.
  35. Subsets of the set of possible states to which
    things converge as time goes to infinity.
  36. These are called attractors.
  37. Attractors, by the way, have a somewhat
    circular definition as what's left after the
  38. transient dies out. There's a way to formalize
    that, which I can put up on our auxiliary video,
  39. if people are interested.
  40. Attracting fixed points are one kind of attractor.
  41. There are three other kinds. We'll talk about
    some of those in the next segment, and all
  42. of them over the course of the next two weeks.
    Now, back to fixed points.
  43. Remember this demonstration? Using the logistic
    map application, that showed that lots of
  44. different initial conditions go to the same
    fixed point.
  45. So if we use the initial condition 0.1, and
    the parameter value 2.2, we go to this
  46. fixed point. Let's try something different.
  47. Different transient, same fixed point.
  48. Different transient, still goes to the same
    fixed point.
  49. The way we think about that behavior, a whole
    bunch of initial conditions going to the same
  50. attractor, is by defining something called
    a basin of attraction.
  51. If you are from the United States, there's an
    easy analogy for you to understand this.
  52. In the middle of the United States, there's
    something called the continental divide.
  53. It runs about ten miles west of where I am
    sitting right now, and a raindrop that falls
  54. to the west of the continental divide will
    run down to the Pacific Ocean.
  55. A raindrop that falls to the east of the
    continental divide will run down to the Atlantic
  56. Ocean, or maybe down Mississippi. and out
    that way.
  57. The analogy here is that the Atlantic Ocean
    as an attractor and the terrain to the east
  58. of the continental divide is the basin of attraction
    of that attractor.
  59. The Pacific Ocean is another attractor, and
    the terrain to the west of the continental
  60. divide is the basin of attraction of that
    attractor, and the boundary of the basin
  61. of attraction divides those two basins.
  62. What do you think will happen to a raindrop that
    falls exactly perfectly on that basin boundary?
  63. Now let's go back and explore what happens
    if we change the R parameter while keeping
  64. x_0 fixed, that is, using the same initial
    condition. There's R=2.3, R=2.4, R=2.5,
  65. as I mentioned in the last segment, the
    fixed point moves. That's like the population
  66. of rabbits stabilizing at a higher number
    if the foxes are less hungry or the rabbit's
  67. birth rate is higher.
  68. Now if you look closely, you'll see that the
    transient lengths differed in that experiment
  69. I just did. R=2.2, the population stabilized
    really quickly. It took a little longer at R=2.3.
  70. The analogy there is that the population
    takes a little bit longer to converge to its
  71. fixed point ratio of foxes and rabbits. You
    also may have noticed, this little overshoot
  72. right here, which gets more pronounced if we raise
    R further.
  73. There's R=2.6, R=2.7, what's going on here
    is that the orbit is still converging to a fixed
  74. point, but instead of converging in a one-
    sided fashion, it's converging in an oscillatory
  75. fashion. It's kind of like, if you push down
    on the hood of your car, and the car bounces
  76. up and down for a while, before settling out.