Dynamical Systems     MATH 374

Homework Assignment #1

Due Wednesday, Sept. 7, START of Class

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.)

  1. In the figures of 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?

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

  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?