Day 22 in Mat115

Today:

• Announcements:
  • Our next exam is Thursday, 11/18.

  • Your voting experience homeworks are due today.

  • Your project revisions are also due today.

  • Election results (so far):
    • the pictures (some pictures won't be there yet, since you may have embedded them in a word doc, or just recently sent them).
    • If you haven't given me
      • County,
      • Precinct info, and
      • exact vote totals,
      I don't have enough from you! Check what you sent me, and make sure that you gave me all of that....
    • What do I do with this?
      • "Out of the twenty members we had participated only one voted for Conway."
      • Please be absolutely specific about the vote! How many for Conway, how many for Paul.

• Today: More Chaos
  • Laplace's notion: that, given the positions and velocities of all the particles in the universe, we should be able to predict the future of "the system" (the universe) exactly for all time. This is the "determinism" in "free will versus determinism". Fortunately for free will, determinism is shown to fail.

  • Chaos is roughly defined as "sensitive dependence on initial conditions", but what this means is that if you start at two different points in a system (for example, in weather you might be thinking of a state of three variables, temperature, barometric pressure, and wind speed), very close to each other but distinct, then eventually the system will produce wildly different behaviors (even those the starting conditions were extremely close).

  • Some classic examples of chaos
    • Hexstat Probability Generator -- out of chaos comes some predictability

    • "The Swinger"
      • the fractal diagram, showing which magnet will result if you start at a given point (each of the five magnets on the pentagon is associated with a color)
      • the path of the pendulum

    • Chaotic Dynamical Systems (and the laws that drive them)

    • The Lorenz Equations
      • Here are the equations which started it all:
      • The Lorenz attractor in 3-D
      • Make your own "butterfly"!

    • Iterated maps:
      • A gorgeous example of what are called "Julia Sets".

      • The logistic map:

        • The recurrence relation:
          

        • This will lead us back to fractals, as well as to the new idea of "Cobwebbing". Let's try a little, using our handout.

        • Here's a nice animation of the process, including the cobwebbing.

        • Excel makes this easy!

        • The "bifurcation diagram". This tells us that, as we vary r, the behavior of the system is different. And at particular values of r, fairly dramatic differences occur.

    • A related topic (complicated dynamics from simple rules): introduction to the Game of Life:
      1. The game is generally played on an infinite checkerboard, or on a world that wraps back around on itself.
      2. The rules are:
         • A living square will remain alive in the next generation if exactly two or three of the adjoining eight squares are alive in this generation; otherwise, it will die.

         • A dead square will come to life if exactly three of its adjoining eight squares are alive; otherwise, it will remain dead.

         • Create your own creatures!
           • Can you think of one that will lead to extinction?
           • Can you think of one that will lead to a stable pattern?

Links:

• Logistic Chaos demonstrator
• www.bitstorm.org/gameoflife/
• www.radicaleye.com/lifepage/
• grid paper
• Way too much detail about the Lorenz Equations