Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to the course text by Devaney. Unless otherwise indicated, all parts of a problem (a), (b), etc. should be completed. You should write up solutions neatly to all problems, making sure to SHOW ALL YOUR WORK. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be YOUR OWN WORK.

** Note: ** Please list the names of any students or faculty who you worked with
on the assignment.

Answer the following questions, writing in
**COMPLETE SENTENCES.**

**Chapter 1 (pp. 1 - 8) **

Read Chapter 1 of the course text, including the wonderful figures.
(As mentioned in class, research student Trevor O'Brien, HC '05, created
pictures similar to Plates 36-38 in his honors thesis work while Gabe Weaver,
HC '04, found figures similar to Plate 39 while working with me one summer on
a HC Fisher grant.)

- In the figures of Julia sets, what do the black points represent? What do the colored points represent?
- When was the Mandelbrot Set first seen and who is it named after?
- According to Devaney, who was the first person to discover "chaos theory"? What problem was he working on and did he actually solve it?
- What role has the computer played in the development of the theory of dynamical systems?

**The Butterfly Effect**

Read the handout *The Butterfly Effect*, the first chapter of
James Gleick's book Chaos.

- Explain what the Butterfly Effect is. Who first discovered it and how?
- Why does the Butterfly Effect make it so difficult to predict the weather?
- What is the technical, mathematical name for the Butterfly Effect? We will see this concept frequently throughout the course.
- The author makes the point that the Butterfly Effect is actually necessary for the Earth's weather system. Explain what he means by this.
- What is another natural phenomenon other than the weather where you might expect to witness the Butterfly Effect?