Homework 3

Homework 3: Due Monday 22 January at the beginning of class

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.

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.

If you want, you may do this set of experiments with a partner, and hand in only one write-up. Think of this as more of a "lab report" than a normal math assignment with a bunch of right answers. I'm mainly interested in seeing that you've explored the system thoroughly and made some reasonably careful observations. This isn't meant to be a stressful ordeal.

Consider the following values for the parameter r:

r = 0.5
r = 1.5
r = 2.8
r = 3.2
r = 3.5
r = 3.56
r = 3.835
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, sketch the general shape of the orbit diagram. (There's no need to print out the graph unless you really want to.)

Finally, pick two other values of the parameter r, between 0 and 4. Report on the long-term behavior of the orbits.

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