Chaos and fractals. An elementary introduction
type: Item
instance of: scholarly article
zbMATH Open document ID: 1252.37001
author: David P. Feldman
publication date: 2012-08-22T00:00:00Z
Mathematics Subject Classification ID: 37-01
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 00A05
zbMATH DE Number: 6069983
zbMATH Keywords: chaos
zbMATH Keywords: fractal geometry
zbMATH Keywords: nonlinear dynamics
zbMATH Keywords: complex dynamics
zbMATH Keywords: cellular automata
MaRDI profile type: Publication
title: Chaos and fractals. An elementary introduction
review text: This is an interesting and unconventional text book aimed at introducing students with modest mathematical background (in North American terminology one might optimistically pitch it at exceptionally energetic students whose major is not in the sciences with mathematical courses prior to but not including `calculus'). This locates the main topic -- dynamical systems, chaos, and insight into the way that simple mathematical models can exhibit complex dynamical behaviour -- much lower down in the mathematical tree than is usual. The preface opens with a well-know quotation from May's influential Nature paper of 1976 arguing that the way in which nonlinear models of the simplest kind lead inevitably to chaotic phenomena should be taught both early (prior to `calculus') and widely (in politics, economics, and so on). This book is one attempt to flesh that out, and it is a very largely successful one. The book is split into seven main parts: an introduction to discrete dynamical systems viewed as iteration of a map, some basic tools for graphing iterates, some early models, and a short detour into the history of debates about determinism and the origins of our belief in mathematical models for the physical universe; `chaos' as observed initially via the logistic equation, Lyapunov exponents, sensitive dependence on initial conditions, bifurcation diagrams, and some statistical language; fractals arising in this context and some fractal geometry, box counting dimension, and more probabilistic language; some complex dynamics and an introduction to the Julia and Fatou set and the idea of parameter spaces, the Mandelbrot set in particular; a quick overview of higher-dimensional and other kinds of systems, notably differential equations and cellular automata; a short narrative overview trying to draw conclusions about the pervasive presence of chaotic phenomena and fractal geometry; and finally three appendices on algebra, statistics, and further reading.NEWLINENEWLINEThere a…
reviewed by: Thomas B. Ward
description: scientific article; zbMATH DE number 6069983