Set-valued Markov chains and negative semitrajectories of discretized dynamical systems
type: Item
instance of: scholarly article
title: Set-valued Markov chains and negative semitrajectories of discretized dynamical systems
zbMATH Open document ID: 1029.37054
author: Phil Diamond
author: Igor G. Vladimirov
published in: Journal of Nonlinear Science
publication date: 2003-10-28T00:00:00Z
review text: Any computer simulation of dynamical systems involves a finite set phase space caused by the nature of finiteness of used machine arithmetic and a rounding to this grid. This paper deals with a mathematical justification of how certain asymptotic distributions arise in computer implementation on models of fixed-point computer arithmetic. It is supposed that the underlying system is a differential equation or an iterated function \(T\) defined on a continuum \(X \subset \mathbb{R}^n\), resulting to its computer implementation by an iterative algorithm, represented by a function \(T_\varepsilon\) defined on the \(\varepsilon\)-grid \(X_\varepsilon = X \cap (\varepsilon \mathbb{Z}^n)\) of \(X\). As we all know, if \(T\) is a smooth invertible function on a bounded open set \(X \subset \mathbb{R}^n\), then its discretization induces a dynamical system on \(X_\varepsilon\) and the discretized function \(T_\varepsilon : X_\varepsilon \to X_\varepsilon\) is neither injective nor surjective. On the one hand, positive semitrajectories can collaps to a common orbit. On the other hand, there might be unreachable points in \(X_\varepsilon\), apart from being initial values themselves. Asymptotic probabilistic properties of iterations of \(T_\varepsilon\) as \(\varepsilon \to 0\) are investigated by considering negative semitrajectories of the discretized system. In contrast to the original invertible \(T\), for further asymptotic studies, one considers associated set-valued Markov chains. In fact, the main theorem of the authors relates the asymptotic behavior of negative semitrajectories of the discretized system to an associated set valued Markov chain. The evolution of these set-valued semitrajectories is studied by the quantization errors of the compensating system generated by discretizing \(T^{-1}\). It is important to emphasize that the considered asymptotic probabilistic laws are not necessarily uniform distributions in general, compared to other texts. The results of t…
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 37M25
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 65G50
Mathematics Subject Classification ID: 65C40
Mathematics Subject Classification ID: 34C28
Mathematics Subject Classification ID: 37C50
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 37E25
zbMATH DE Number: 1931216
zbMATH Keywords: spatially discretized dynamical systems
zbMATH Keywords: finite machine arithmetic
zbMATH Keywords: iterated function systems
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: set-valued Markov chains
zbMATH Keywords: semitrajectories
zbMATH Keywords: compensating system
zbMATH Keywords: quantization errors
MaRDI profile type: Publication
full work available at URL: s00332-001-0450-4
OpenAlex ID: W2153676434
DOI: 10.1007/S00332-001-0450-4
description: scientific article; zbMATH DE number 1931216