In the 2005-2006 school year, Augsburg senior math majors Dan Wolf and Melissa Larson worked with Dr. Jody Sorensen on a research project in the field of Dynamical Systems. Dan and Missy had no background in this field, so they started by working quickly through a textbook on the subject. Dynamical Systems involves iterating functions: you start with an x, apply a function f and then take that output and put it back into the function over and over again. The goal is to be able to predict what happens in the long term. Dynamical Systems has practical applications ranging from economic modeling to population dynamics to weather prediction.

Dan and Missy’s work focused on the bifurcation diagram, an example of which is shown below. The example is for the function f(x) = x2+c, and has the parameter c on the horizontal axis and the variable x on the vertical axis. The picture shows the periodic points for various c values - these are the points that are locked into a repeating pattern under iteration. The colors vary based on how those periodic points effect nearby points - whether they are attracting or repelling.

In looking at many examples of bifurcation diagram, Dan and Missy and Jody noticed certain common traits. For example, near places where the diagram branches, the periodic points always alternate between attracting and repelling. They were able to prove (under very general conditions) that this alternating behavior must occur. Surprisingly, the proofs ended up using the Intermediate Value Theorem, a topic covered early on in every Calculus I class.