APPLICATIONS OF MATHEMATICS – MATH03-400-01
Summer/Fall, 2007
Dr. Marcus Wright – wright@rowan.edu
Office Hours: M – R 12:30 – 1:30
856-256-4500 ext. 3873
Math. Dept. Office: 256-4844
INTRODUCTION TO CHAOTIC DYNAMICAL SYSTEMS
Theory and Applications
PREREQUSITES: Calculus III and Linear Algebra
TEXT: CHAOS – An Introduction to Dynamical Systems, by Kathleen Alligood, Tim Sauer and James Yorke. Springer ISBN 978-0-94677-1
Other supplements will be supplied.
Mathematica, graphing calculators and other technologies will be used.
Students who know a computer programming can use this as well for some assignments.
About the course: Introducing students of mathematics to the topics we will study at the undergraduate level by offering the kind of course we will have is a “contemporary” phenomenon. We are going to study and try to understand deeply some examples of nonlinear dynamical systems, beginning with (real) one-dimensional examples of discrete dynamical systems. The advantage of studying dimension one is that we have easy to comprehend graphical and computational methods at our disposal. Although the systems are discrete (i.e., sequential), many of the basic theorems of continuous and differential mathematics, i.e., the calculus, will be employed. This course will truly be an “excursion in the uses of calculus”. Linear algebra will also have an important place in our toolbox.
In Chapter 2 will branch out to study higher dimensional examples – we will see how Poincare used a two dimensional dynamical system to study the motion of systems of particles in three dimensions and more. We will see how to use the theory of linear maps to classify fixed points of nonlinear map.
In Chapter 3 we will study definitions of chaos, and quantitative measures of chaos, such as the Lyapunov exponent. We will go back to one dimensional examples for some computations. Then in Chapter 4 fractals will enter the picture. In Chapter 5 we will round out our investigation into discrete dynamical systems by looking at the horseshoe map.
The course will conclude with some topics about chaos in differential equations, including one that started a lot of modern interest in the subject, the Lorenz attractor. These will come from Chapters 7 through 9.
Research Project
Each of you will be expected to pick a topic not covered to learn-research. And you will be asked to give a presentation of what you have learned to the class at the end of the semester, together with a written version. I have lots of suggestions about topics, but I bet some searching of your own will discover many more. GET STARTED ON THIS EARLY. I AM ALWAYS WILLING TO DISCUSS THIS PROJECT WITH YOU.
GRADING
Class Participation ≥5%
Homework: 15% (Can be collaborative, but each
student will have turn own version}
Midterm Exam 30%
Final Exam 30%
Learn-Research 20%
NOTE: I would like to have students contribute to the learning of the class in as many ways as possible, including showing us examples from you major, demonstrating applets or other things you found on the web, and giving unusual ways to do problems, etc.
Assignment for Thursday and Tuesday, September 3 and 8:
1. Read the Introduction, paying close attention to the section Background. Here you will get a little bit of information about the history and nature of the subject.
2. Read pages 1-24. Don’t worry too much about all the details, but make sure you understand the definitions and could reproduce them.
3. Hand in exercises T1.2, E1.1 next Tuesday.
4. Checkout E1.2 and E1.3. (Checkout means you should try to do and if not successful, come to class with questions about.)
NOTE: We will be going over this material for about a week.