Borderline Behavior for 2 x 2 Iterative Systems

Abstract

The study of $2 \times 2$ linear iterative systems leads naturally to an analysis of the eigenvalues and eigenvectors of the corresponding system matrix. The phase portraits for such systems have been previously examined and outlined; however the outline lacks the analysis of the many borderline cases in the trace-determinant plane. In this paper we fill in some of these details and look at the general solutions for the most interesting cases in terms of eigenvectors. In particular, we find generalized eigenvectors when required.