Effects of phase space discretization on the long-time behavior of dynamical systems
type: Item
instance of: scholarly article
title: Effects of phase space discretization on the long-time behavior of dynamical systems
zbMATH Open document ID: 0617.65066
DOI: 10.1016/0167-2789(87)90100-X
author: Christian Beck
author: Gert Roepstorff
published in: Physica D
publication date: 1987-01-01T00:00:00Z
review text: The effects of the phase space discretizations and of the roundoff errors on the period, L, of the loops of the iterates \(f^ n(x_ 0)=f(f^{n- 1}(x_ 0))\), \(n\geq 1\), of a given function f(.): I\(=[-1,1]\to R\), \(x_ 0\in I\), are investigated through a large number of computer experiments with functions of the form: \(f_{\mu}(x)=1-\mu | x|^ z\). Dividing the interval \(I=[-1,1]\) into N cells of size \(\Delta\) and plotting log L against log N for \(\Delta\to 0\) the authors conclude that the period L of the loops of the iterates \(f^ n(x_ 0)\), \(n\geq 1\), \(x_ 0\in I\), satisfies a relation of the form: \(L\leq N^{\epsilon}\) where \(\epsilon\) is empirically guessed. For ergodic discrete dynamical systems represented by the system defined by the function \(f(x)=1-2x^ 2\) as well as for ''critical systems'' defined by functions of the form: \(f(x)=1-\mu_ cx^ z\) where \(\mu_ c=\mu_ c(z)\) is a critical value, some theoretical arguments to explain the numerical results are given.
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 65Q05
Mathematics Subject Classification ID: 93C55
zbMATH DE Number: 4001353
zbMATH Keywords: roundoff error effects
zbMATH Keywords: periodic orbits
zbMATH Keywords: numerical experiments
zbMATH Keywords: ergodic discrete dynamical systems
zbMATH Keywords: critical systems
zbMATH Keywords: critical value
zbMATH Keywords: numerical results
reviewed by: Ştefan Mirică
MaRDI profile type: Publication
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full work available at URL: 0167-2789(87)90100-x
OpenAlex ID: W2046402891
description: scientific article; zbMATH DE number 4001353