Dynamical Systems     MATH 374

Prof. Gareth Roberts

Homework Assignment #1

Due Thursday, September 7, START of Class

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.

1. Explain what the Butterfly Effect is. Who first discovered it and how?
2. Why does the Butterfly Effect make it so difficult to predict the weather?
3. What is the technical, mathematical name for the Butterfly Effect? We will see this concept frequently throughout the course.
4. The author makes the point that the Butterfly Effect is actually necessary for the Earth's weather system. Explain what he means by this.
5. 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 visible at the top of the course webpage.)

1. In the figures of the various Julia sets, what do the black points represent? What do the colored points represent?
2. When was the Mandelbrot Set first seen and who is it named after?
3. According to Devaney, who was the first person to discover "chaos theory?" What problem was he working on and did he actually solve it?
4. 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 D2(x) and D3(x) as piecewise functions.

Additional Problem: Let f(x) be a continuous function whose domain and range is all real numbers. Suppose that the orbit of x0 under f converges to the point a, that is, x0 is on an asymptotic orbit. Using the definition of continuity, prove that a is a fixed point of f.