Homework 5


Homework 5: Due Friday 22 April, 2005. As on the previous computer assignment, you can work in pairs and hand in only one set of answers if you want.

If you have questions about any of these, be sure to ask me. These aren't intended to be tricky or vexing. If you have computer difficulties, let me know.

To answer these questions, you'll need to use a program to make bifurcation diagrams. There are many such programs on the web. Here are some to choose from

You will also want 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.

And lastly you'll want the program I used in class to plot simultaneously the orbits for two different initial conditions. This program can be found at: http://hornacek.coa.edu/dave/Chaos/initial.conditions.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.)
    You may wish to confirm your findings by plotting the time series for the parameter value using the orbit plotter program from last assignment.

  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. 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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