Introduction to Chaos and Fractals

Homework 5

Due Friday 21 April, 2006

If you want to do these exercises in pairs and hand in only one write-up, that's fine with me.

Chapter 8
  1. 1
  2. 3
  3. 4
  4. 6 (optional. do this if you feel like doing some algebra)
Bifurcation Diagram Explorations 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. A good web-based program is http://www.cs.laurentian.ca/badams/LogBig/LogBigApplet.html.

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.
    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. For each r value, do the following. 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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