(in)consistency: Some Low-dimensional Examples

This note is a slightly revised version of an earlier note with the same title that was part of a larger research project of the Dynamical Systems Group at Ohio University. While the major findings of the project are described in [1], this note complements [1] as it contains a more extensive review of some basic low-dimensional examples of Boolean systems and their ODE counterparts and explores whether the ODE dynamics is consistent with the Boolean dynamics. Date: August 10, 2013; Source: LowDimEx18.tex 1. Notation and some basic definitions Our notation roughly follows the one in [1], but as the original version of this note substantially predates [1], there are some important differences. Since the original version of this note forms part of a larger body of interconnected notes that contain additional unpublished results of independent interest, we will retain here the original terminology. Therefore let us briefly review some basic definitions and frequently used notations and display the differences between the terminology used here and the one of [1] in the form of remarks. Throughout this note, we let f denote the updating function of an ndimensional Boolean system. We will let d(x, y) denote the Euclidean distance between vectors x, y ∈ Rn. We consider the following functions on the set of reals: g(xi) = 3xi − xi − 3 S(xi) =  0 if xi ≤ −1, .5(xi + 1) if −1 < xi < 1, 1 if xi ≥ 1 . s(xi) = { 0 if xi ≤ 0, 1 if xi > 0.