A first course in discrete dynamical systems
type: Item
instance of: scholarly article
title: A first course in discrete dynamical systems
zbMATH Open document ID: 0797.58001
author: R. A. Holmgren
published in: Universitext
publication date: 1994-05-26T00:00:00Z
review text: Discrete dynamical systems are essentially iterated functions. The book under review is suitable for a one-semester course on discrete dynamical systems. It contains an extensive quantity of interesting exercises. Let us specify its substance by chapters: the concept of the function; topology of the real numbers; periodical points and stable sets at the iteration of real functions, graphical analysis; Sharkovskij's theorem; dynamical information contained in the derivative of the function, attracting and repelling points; parametrized families of functions and bifurcations (the definition of the bifurcations of parametrized families of functions, bifurcation diagrams, saddle-node, pitchfork, transcritical and period doubling bifurcations); symbolic dynamics and chaos; the logistic function \(h_ r(x) = rx(1 - x)\), \(r > 0\); Newton's method from the point of view of discrete dynamics, the iterations for quadratic and cubic functions; numerical solution of differential equations, Euler's method; the dynamics of complex functions, dynamics of the quadratic maps, Newton's method in the complex plane; Mandelbrot and Julia sets. The Appendix is devoted to computer algorithms (iterating functions, graphical analysis, bifurcation diagrams, Julia and Mandelbrot sets, stable sets of Newton's method).
Mathematics Subject Classification ID: 58-01
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37B99
zbMATH DE Number: 575713
zbMATH Keywords: logistic function
zbMATH Keywords: iterated functions
zbMATH Keywords: discrete dynamical systems
zbMATH Keywords: periodical points
zbMATH Keywords: stable sets
zbMATH Keywords: Sharkovskij's theorem
zbMATH Keywords: parametrized families of functions
zbMATH Keywords: bifurcations
zbMATH Keywords: symbolic dynamics
zbMATH Keywords: chaos
zbMATH Keywords: Newton's method
zbMATH Keywords: Julia sets
zbMATH Keywords: Mandelbrot sets
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 575713