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Homework 3 |
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Homework 3: Due Thursday 15 April 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. 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.)
[Dave]
[Chaos and Fractals]
[Homework Page]
[COA]
Web page maintained by dave@hornacek.coa.edu.