Homework 2

Homework 2: Due Friday September 24?

Background Info

• There are two websites that you'll want to use to help you with this assignment.
  1. Java Applets from the Computational Beauty of Nature by G.W. Flake. I recommend choosing the large screen applets. If performance is slow, however, you may wish to opt for a smaller screen applet. Once you click on "large screen", it may take a few minutes for the applets to load themselves onto your computer. Once the applets are loaded, everything should be pretty quick.

     Once the applets load there's a click-able menu that will appear in the upper right. The three applets you'll want to check out are: One-dimensional time series, one-dimensional phase space, and one-dimensional bifurcation. The 1d phase space applets produces "cobweb diagrams".

     Important note: Flake uses a different convention for the logistic map than most everybody else. His r is 1/4 the value of our r. So, an r=4 for us would be the same as an r=1 for Flake.

     On the bifurcation diagram applet, you can zoom in on a region of the applet by highlighting the applet with your mouse, and then clicking on restart.

     Feel free to play around with some of the other applets. These programs could be great tools to have at your disposal when working on projects. A detailed discussion of the programs is found in the book, The Computational Beauty of Nature, on reserve in the library.

     
     

  2. Last year I wrote a few programs and built a web interface to them. They can be found here. Select the initial conditions link to test for sensitive dependence on initial conditions. The other program, as the name says, lets you make bifurcation diagrams.

     Important Note: After you run the program, please don't hit the "back" key on your browser. Instead, go to the bottom of the page and select "start over". When you run the programs, a temporary data file is stored on my computer. If you click on "start over" then this data file gets erased. If you hit back, the data file doesn't get erased. It's possible (although unlikely) that all these temporary data files could accumulate and fill up my hard drive.

• If you want, you can do this assignment in pairs. (Trios or larger are not ok, however.) If you decide to work in pairs, hand in only one assignment. Each member of the duo will receive the same score/evaluation.

• This assignment is not intended to be terribly difficult or time-consuming. I mainly want you to play around with the logistic map some and gain a greater appreciation for all the things it can do.

• All problems refer to the logistic equation, f(x)= rx(1-x).

The Problems

1. Logistic Explorations. Consider the following parameter values:
   1. 3.594
   2. 3.835
   3. 3.567
   4. 3.7
   5. 3.5699456718695445
   For each of the parameter values, describe the behavior of iterates of the logistic map. Is the map periodic or chaotic or something else? If it's chaotic, justify this carefully by arguing that the map satisfies the criteria listed in section 1.7 of our text. You do not need to include tons of printouts in your write-up. (Although if you think they would be especially helpful, feel free to include them.) However, you should make it clear how and why you've reached the conclusions that you've reached. Don't worry too much if the last r value yields confusing results.

2. Period Doubling.
   1. Determine the range of parameter values for which the logistic equation is periodic with prime period
      1. 3
      2. 6
      3. 12
      4. 24
      5. 48
   2. Use these values to estimate Feigenbaum's number, as defined on p. 30 of our book.

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