Multi-Parameter Functions in Chaotic Dynamical Systems

Abstract

For two semesters, a fellow math major and I thoroughly proved results from Sections 1.1 – 1.8 of An Introduction to Chaotic Dynamical Systems by Robert Devaney. After going through Devaney's calculations and proofs, I created a multi-parameter family of functions to consider and observe. This is a piecewise function of polynomials that always intersects the x-axis at 0 and 1. It has two maxima and one minimum value. Depending on the range of the parameters, the minimum value can be above or below the x-axis. I have analyzed its behavior and determined the fixed and periodic points. I found that at certain parameter values the family of function's corresponding invariant set will be closed and totally disconnected. I conjecture that the invariant set is also a perfect subset of the unit interval which would make it a Cantor set. Next, I conjecture if the same parameter values could be used to show the new equation maps are chaotic. Dr. Brian Raines will guide me through the steps of this process.