The field of dynamical systems is the study of the time evolution
of systems. Time evolution can be continuous (differential equations)
or discrete (repeatedly apply the function that moves the system from
its current state to the state one time period later.) For example,
one might hope to predict the course of the economy given data about
its current state. Or one could hope to predict the weather given its
current status.
Dynamical systems have been studied for many years, but there was a great boom in interest in the 1980's, when computers became powerful enough to visualize dynamical systems with color graphics. This continues to be an area of mathematics where computer experimentation can lend insight, and computers will be used in this course.
Imagine that x represents the current state of a system (e.g. the size of the raccoon population), and f(x) gives the population one year later. We are interested in what happens when we repeatedly apply ("iterate") the function f. We would like to know whether there are fixed points (population sizes that would stay unchanged), or periodic points (will the population oscillate between two or more sizes?) We would also like to know if fixed points are stable (will the raccoons die out, tend toward a stable size, or will the population of raccoons explode?)
We will look at fixed and periodic points and limiting behavior of iterates first for real valued maps. The lowly quadratic map f(x) = x^2 + c has quite interesting behavior. For the right values of c, these maps illustrate chaotic behavior. We'll look at Feigenbaum's famous constant that describes the "period doubling approach to chaos". This shows the surprising fact that a broad class of functions approach chaotic behavior in the same fashion that quadratic maps do.
Fractals play a key role in dynamical systems. A fractal is a figure that is self-similar: magnified views look the same as the original figure. Fractals have the interesting property that they may have fractional dimension. Some well-known examples that arise from iterative processes are the Cantor set, the Sierpinski triangle, and Koch snowflakes. One commercial application of dynamical systems is the use of "iterated function systems" to compress pictures. Fractals also arise from the "chaos game", described in a colloquium talk a few years ago.
Another portion
of the course will deal with complex valued functions. Analysis of
iteration of such functions lead to "Julia sets" and to the
Mandelbrot set. These are the source of many spectacular color
graphics.
The prerequisite for Math 349 is Math 302. The
course applies some techniques of analysis from 302, but is mostly
self-contained.
Math 349 counts toward the mathematics major/minor as a 300-level
elective.
Prerequisite: 302
Distribution: Mathematical Modeling. Majors can fulfill the
major
presentation requirement in this course in 2005-06.
Previous topics have included Graph Theory, Chaotic Dynamical
Systems, Mathematical Logic, and Number Theory
