We will look at fixed and periodic points and limiting behavior of iterates first for real valued maps. For the right values of c , these maps illustrate chaotic behavior. This shows the surprising fact that a broad class of functions approach chaotic behavior in the same fashion that quadratic maps do. One nice feature of dynamical systems is that students can read and participate in current mathematical research. For example, Newton's method involves iteration of functions, and in the past one student's summer research involved convergence of Newton's method.
Fractals play a key role in dynamical systems.
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. Another portion of the course will deal with complex valued functions. These are the source of many spectacular color graphics. The prerequisite for this course is Math 116, 116Z, or 120.