Discrete dynamical systems
type: Item
instance of: scholarly article
title: Discrete dynamical systems
zbMATH Open document ID: 0843.93005
author: François Robert
published in: Mathématiques \& Applications (Berlin)
publication date: 1995-10-08T00:00:00Z
review text: The French Society for Applied and Industrial Mathematics (SMAI) is publishing a series of books, each of which is usually based on courses of lectures given by the respective authors. While the content may be fairly new, each book is therefore intended to have a pedagogical function. This book is one of that series, and carries out its aims admirably. In 1986, an earlier version appeared in English (``Discrete iterations: a metric study'') also published by Springer. It has been revised and completed to form Chapters 2-7 and 10 of the present work (which has five other chapters as well). The author motivates the mathematics by imagining a finite set of \(n\) processors (automata, neurones, \dots) at the vertices of a directed graph \(\Gamma\), such that vertex \(i\) can be in one of two states, and produces output \(f_i (x_1, \dots, x_n)\) which depends only on the input-states received from the edges ending at \(i\). Thus we obtain a mapping \(F\) from the Cartesian product \(X= \{0, 1\}^n\) to itself. The incidence-matrix of \(\Gamma\) yields the Boolean matrix \(B(F)\), which gives crude information about \(F\). Each element of \(X\) is a vector \(x= (x_1, \dots, x_n)\), thought of as a vertex of the unit cube in \(n\)-space, and we form the (directed) iteration graph \(I\) of \(F\) by joining \(x\) to \(y\) whenever \(y= F(x)\). This graph then consists of several connected components, each invariant under \(F\), and will contain either a unique fixed point, or a cycle, to which any orbit of iterates of \(F\) will terminate after a finite number of steps. The object is to decide the structure of the orbits by means of economical algorithms, and the author provides a theory which resembles the standard ``continuous'' theory of metrics, norms and contractions; but the proofs are simpler and all the relevant objects are finitely computable. He also considers various analogues of Gauss-Seidel (``series-parallel'') iteration to compare speeds of approach to a steady…
reviewed by: H.B.Griffiths
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 94C15
Mathematics Subject Classification ID: 94C10
Mathematics Subject Classification ID: 92B20
zbMATH DE Number: 803183
zbMATH Keywords: Hopfield neuronal network
zbMATH Keywords: discrete iterations
zbMATH Keywords: directed graph
zbMATH Keywords: Boolean matrix
zbMATH Keywords: cycle
zbMATH Keywords: metric
zbMATH Keywords: iterations
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 803183