Homework Four


Due at the start of class, Monday 25, 2009.

For all these problems, use the program at http://hornacek.coa.edu/dave/Chaos/orbits.html. All of the questions below concern the logistic equation, f(x) = rx(1-x). You will explore what happens to orbits of f(x) for different values of the growth rate r, as we did in class on Thursday.

If you want, you may do this set of experiments with someone else in the class, and hand in only one write-up. I'm mainly interested in seeing that you've explored the system thoroughly and made some semi-careful observations. This isn't meant to be a stressful ordeal. Consider the following values for the parameter r:

  1. r = 0.8
  2. r = 1.8
  3. r = 2.9
  4. r = 3.2
  5. r = 3.5
  6. r = 3.56
  7. r = 3.835
  8. r = 4.00

For each of the parameter values, observe what happens to orbits of f(x). Consider only initial populations between 0 and 1. For each parameter value, you should:

  • Determine the long-term behavior of the system. Does the population die off, reach a fixed point, or reach a periodic point? For some parameter values you might need to plot a few thousand orbits to see the final behavior.
  • Try a few different initial conditions for each parameter value. Remember that your initial conditions should always be between 0 and 1. You should find that overall behavior is independent of the initial conditions you choose.
  • For each parameter value, make a rough sketch the general shape of the time series plot. (There's no need to print out the graph unless you really want to.)