Introduction to Chaos and Fractals

Project.

Please write a brief update on your ideas for a project topic. In particular, please let me know what references, if any, you've found. Also, let me know if you need help finding references or developing your topic further. This isn't a binding proposal and it needn't be a formal write-up. I'm just looking for a quick update, either via email or paper.

Bifurcation Diagram Explorations.

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.)
2. For each r value, do the following.
   • Determine the long-term behavior of the orbits. Are the orbits periodic (what period?) or chaotic?
   • Does the equation show the butterfly effect?
   1. 3.7
   2. 3.835
   3. 3.5699456718695445 (don't round off).