Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to the primary course text by Robert 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 your work. A nonempty subset will be graded. You are encouraged to work on these problems with other classmates, and it is ok to use internet sources for help if it's absolutely necessary; however, the solutions you turn in should be YOUR OWN WORK and written in YOUR OWN WORDS.

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

**The Butterfly Effect**

Read *The Butterfly Effect*, the first chapter of
Gleick's book on Chaos. Answer the following questions, making sure to write in
COMPLETE SENTENCES.

- 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?

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

Read Chapter 1 of Devaney's textbook, making sure to check out the wonderful figures.
(Side note: my former 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 as part of the
HC Summer Research program. A figure created by each student is presented at the top of the
course webpage.) Answer the following questions, making sure to write in complete sentences.

- In the figures of the various 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?

**Chapter 3 (pp. 26 - 28) **

Problems: 3, 7 (b, c, e, h), 11, 12, 13, 14

**Note:** For problem #12, you should give explicit formulas for **D ^{2}(x)** and

**Additional Problem:** Let **f(x)** be a continuous function whose domain and range is all real numbers.
Suppose that the orbit of **x _{0}** under