Homework 3


Homework 3: Due Friday 28 January

Problems from Devaney

  1. Exercise 44, p. 22. Be sure to consider what happens to all possible initial conditions.
  2. Exercise 3.1, p. 43.

Computer Problems. Please be sure to do these before class on Friday!!! For all these problems, you will probably want to 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. 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 thoughtful and careful observations.
  1. For all of the following values of r, figure out as best you can what happens to iterates of f(x) with initial conditions between 0 and 1.
    1. r = 0.5
    2. r = 1.5
    3. r = 2.5
    4. r = 3.2
    5. r = 3.49
    6. r = 3.55
    7. r = 3.83
    8. r = 4.00
  2. For all of the above parameters discuss the behavior you see for initial conditions between 0 and 1. Is the system periodic? Can you concoct a simple (one or two sentence) story about the population of rabbits on an island?


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