An investigation

This isn't an official homework; there's no need for a fancy write-up. But you should have some notes to hand in. I'm not going to evaluate this, I just want to see what you've found and check off that you did it. I expect that doing these will take around half an hour.

For all these problems, you will probably want to use the program at http://hornacek.coa.edu/dave/Chaos/orbits.html. If you prefer, you could use a spreadsheet program or a programmable calculator or FORTRAN or an abacus.

All of the questions below concern the logistic equation, f(x) = rx(1-x). You will explore what happens when you iterate of f(x), for different values of the growth rate r.

1. Consider the following values for the parameter r:
   1. r = 0.5
   2. r = 1.5
   3. r = 2.8
   4. r = 3.2
   5. r = 3.5
   6. r = 3.55
   7. r = 3.835
   8. r = 4.00
   For each of the parameter values, observe what happens when you iterate f(x). Consider only initial populations between 0 and 1. For each parameter value, you should:
   • Determine the long-term behavior of the system. What happens to our mythical shark population? 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, sketch the general shape of the population as a function of time. (There's no need to print out the graph unless you really want to.)

2. Pick two other values of the parameter r, between 3 and 4. Report on the long-term behavior of the orbits.

Web page maintained by dave@hornacek.coa.edu.