Math W80
Chaos, Nonlinear Dynamics, and Applications
Interim, 2003



Welcome to a course on chaos and nonlinear dynamics, an area of mathematics which has only been explored in depth since the advent of high-speed computing, and one of much current research interest. In previous courses we have learned the processes of differentiation and integration and, particularly for those who have taken Math 231, used these to model and predict future behavior in physical processes. In this course we will encounter the effects of nonlinearity upon such models — particularly, the implications of nonlinearity upon predictability. We will see how tweaking parameters in a model can lead to radical behavioral changes (bifurcations), how a sequence of such bifurcations often leads to a chaotic system, and how chaotic trajectories still yield a hidden order to them, being drawn to a strange attractor (generally a fractal).

Class Meetings

Our class meets Mon. — Fri. 8:30—11:30 am, with a break for chapel. Our classroom, SB 372, is available to us both during this time period, and during the afternoons. You should plan to spend a significant amount of time in this (or some other Mathematica-equipped) room besides the regular class period, as many of the assignments will require a computer.

Contacting the Professor

Generally speaking, one (but not both) of Professors Wagstrom and Scofield will be available for help in the afternoons. The one of us that is “on call” will generally be in his/her office (Wagstrom NH 286; Scofield NH 281) or in the classroom. The other is likely to be in his/her office preparing materials for future class sessions; please resist the urge to interrupt. With a couple of exceptions, the “on call” professor will be the one who conducted class on that day.

Use of Technology

A good deal of computer use will be expected of you in this course. Generally speaking, daily homework assignments are available on the the web, and it is your responsibility to visit the homework page to find out what they are. While many announcements, hints, etc. may be given in class, things that cannot wait until the next class period will be sent to you as email messages. Thus, it is important that you be checking your email at least once at the end of the day. We have requested a class email list to which you can send messages at

Any mail sent to this address will be received by all members of the class (including the professors). You may use it as a forum for discussing assigned problems, topics that came up in class, etc.

In addition, a number of assigned problems will require the use of software. While this will often be Mathematica, results obtained via some other means (software you have written, another symbolic manipulator like Maple or MathCAD, even a java applet) are usually permissible if well-documented with explanations of the commands used and the output received.


As in most Interim courses, no letter grades will be assigned; “grades” will be H (honors), S (satisfactory) or U (unsatisfactory). Your performance on daily assignments and the final group project will play the major role in the determination of your “grade”.

Daily Assignments (10; 50% of grade)

On most days, an assignment will be made to be worked on that afternoon and evening and submitted the next day. The assignment will generally consist of two types of problems.

Exams (2; each 10% of grade)

These will be given out much like the daily assignments discussed above, with answers due at the beginning of class on the next day. The major differences between these and daily assignments are that

Group Project (30% of grade)

As you work with various classmates on daily assignments, keep an eye out for people who have interests in common with you in preparation for forming a 3-person group. Also, pay attention to possible project topic suggestions as they are made in the reading or by the instructors. Toward the end of the term you will be given a couple of days of class time to research these topics, pool your knowledge together with your fellow group members, and prepare both a talk on the subject (to be presented jointly to instructors alone) and a poster (presented to the full class).


Earning an “honors” grade in the course requires that you receive an “H” on four individually-submitted exploratory problems (according to the rules indicated above, you may submit as many as 7 of these to be graded), earn at least 86% on both exams, and do reasonably well on your group project.

We consider the distinction between exploratory problems worked in a group and those worked individually an important one. When collaboration has occurred between 2-3 people, indicate this by submitting one problem solution for the group, providing all names of the group members (never more than 3). There are a number of reasons experience working in groups is important, none of which will be mentioned here. But problems submitted by an individual should definitely constitute more than an individual write-up to a collaborative process. The work and ideas should substantially be your own. Two students who say they are working individually, yet draw non-obvious conclusions which are substantially the same, should not expect to receive an “H” on their write-ups.

This page maintained by: Thomas L. Scofield
Department of Mathematics and Statistics, Calvin College

Last Modified: Friday, 31-Jan-2003 09:54:47 EST