Homework 5


Homework 5: Due Friday 23 April, 2004. As on the previous computer assignment, you can work in pairs and hand in only one set of answers if you want.

To answer these questions, you'll need to use a program to make bifurcation diagrams. Such a program can be found at http://hornacek.coa.edu/dave/Chaos/bifurcation.html. If you don't like my program or it's running too slow or you just want to try something different, you could check out a java applet that's available here on the web page of Bob Devaney's research group. The java applet basically does the same thing as the program on my website. If you have a slow connection but a reasonably fast computer, the applet may be the choice for you.

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 the partition of my hard drive that I've allocated for these web programs.

You may also wish to use the program from last homework to calculate orbits of the logistic equation. This program can be found at http://hornacek.coa.edu/dave/Chaos/orbits.html.


  1. By experimenting with the bifurcation diagram program, find r values that yield orbits with the following properties. Once you're found the r value, check that it's behaving as you expect by using the orbit program. There are many possible answers to these questions. Briefly summarize your findings. You don't need to print out any graphs, unless you find some that look really neat or are particularly helpful for explaining things. The last two might be a little challenging. Give them a try, but don't worry if you can't find them.
    1. Period 4
    2. Period 6 (Hint: Look near period 3.)
    3. Chaotic behavior for some r not equal to 4. (There are many possible r values to choose from.)
    4. Period 5 (Hint: Look between 3.7 and 3.8.)
    5. Periodic behavior of some other period that's not a multiple of 2. (Be sure to state what the period is you've found.

  2. On the bifurcation diagram you should notice a bunch of structures that look like sideways pitchforks. Zoom in on a few of them. What do you find? (Just describe the what you see qualitatively -- this isn't a technical question.)

  3. By zooming in on the bifurcation diagram, estimate the "accumulation point". This the parameter value at which the period doubling behavior "accumulates" and turns into chaos. For a description of the accumulation point, see the top of page 73 of the Gleick book. It will probably be difficult to get more than two decimal points accurate.

  4. For these problems you'll want to use the three computer programs located at http://hornacek.coa.edu/dave/Chaos/initial.conditions.html. For each r value, do the following. A.Determine the long-term behavior of the orbits. Are the orbits periodic (what period?) or chaotic? B.Does the equation show the butterfly effect? Sketch or print out any graphs you use to draw your conclusions.
    1. 3.7
    2. 3.835
    3. 3.5699456718695445 (don't round off).


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