The Mathematics of Chaos
Instructor: Suzanne Sumner
Department: Mathematics
Course Number: FSEM 100D
CRN: 12735 or 12736
Course Overview: Chaos theory is dependent on the continuum of the real numbers. The main purpose of this course is to explore the real number system and higher-dimensional Euclidean space in much more detail than a traditional calculus course allows, while focusing on concepts that require no formal background in calculus. This will be accomplished mainly through the study of discrete dynamical systems. Over the course of the semester you will:
- explore the concept of a function beyond what is typically taught in Algebra or Calculus
- learn the process of function iteration
- develop a deeper understanding of the real numbers
- understand the concepts of limits and continuity as they relate to functions
- learn the concept of a metric space
- study some famous results from the genesis of chaos theory
- learn some properties of complex numbers and complex functions
- explore the concept of Julia sets and the Mandelbrot set
- develop an understanding of fractals in Euclidean space
- work with applications using discrete dynamical systems
- learn how to use the software package Dynamics Solver
- develop skills to approach mathematical concepts and problems with greater depth
- develop writing skills both general and specific to the discipline of mathematics
Class Materials:
- Some Mathematical Models with Very Complicated Dynamics by R. May
- Period 3 Implies Chaos by T. Li and J. Yorke
- Chaotic Dynamics in an Insect Population by R. Costantino et. al.
- selections from Chaos by Alligood, Sauer, and Yorke
- As there is no formal text for this course, your notes will be a critical resource. Make sure that you have access to a fellow student's notes in the unlikely event that you miss a class.
- A TI graphing calculator is required for this course. Any version 82 or higher is fine; I will be using an 83 for in-class presentations.
- You will be required to use a software package called Dynamics Solver, which can be downloaded free of charge at: http://tp.lc.ehu.es/jma/ds/ds.html The current version is 1.70 and works on Windows XP. Dynamics Solver is also available on the computers in the Trinkle labs. You will be expected to complete written assignments in a word processor. If you choose to use Microsoft Word you will need access to the built-in Equation Editor for this program and will be expected to learn how to use it.
Grading: Grades will be based mainly on formal writing assignments throughout the semester. There will be five written assignments in which you will incorporate output from Dynamics Solver into a document with thorough explanations of the underlying concepts and relevant examples. These will be worth 10 points each; deadlines will be given with each assignment and late assignments will lose two points for every calendar day overdue. These will be graded on both content and style, so you should be prepared to learn some form of mathematical word-processing software such as the Equation Editor package included with Microsoft Word.
There will be a final project with a presentation during the last week of classes and a paper due Monday, December 3rd, which will focus on a topic of your choice. Your idea should fit into one of these three categories:
- a formal research project on some historical topic or scientific field relevant to chaos theory, with a heavy emphasis on primary sources;
- a study of a specific application of dynamical systems;
- a computer exercise with an emphasis on programming and graphics.
You will submit a rough draft of your final paper on Monday, November 19th worth 5 points; the final paper will be worth 10 points and the presentation also worth 10 points.
There will also be a cumulative, take-home final exam worth 25 points that will cover mainly definitions, theorem statements and some simple computations. Though not given formal weight, attendance and class participation will be expected.