Introduction to Chaos and Fractals

Homework 4

Do these exercises before class on Monday January 21, 2008

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. 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:

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, make a rough sketch the general shape of the orbit diagram. (There's no need to print out the graph unless you really want to.)