Just Tilt Your Head: A Graphical Technique for Classifying Fixed Points

Abstract

Summary Iterations of the form are used throughout mathematics, and convergence (or not) of an iteration to a fixed point is always a central question. We present an informal graphical technique for resolving whether a fixed point in one dimension is attracting or repelling. Specifically, given a real-valued function f, we are determining whether or at a point for which . The former condition on classifies the fixed point as attracting, and the iteration as locally convergent; the latter corresponds to a repulsive fixed point, in which the iterations get further away from the fixed point no matter how close you start them off. The technique requires only that we be able to compute points on the graph of and not necessarily on an explicit functional form or its derivative.