Rigorously computed orbits of dynamical systems without the wrapping effect
type: Item
instance of: scholarly article
title: Rigorously computed orbits of dynamical systems without the wrapping effect
zbMATH Open document ID: 0910.65052
DOI: 10.1007/BF02684450
author: K. Appert
published in: Computing
publication date: 1999-04-09T00:00:00Z
review text: Consider the discrete dynamic system \[ \Omega_n=f_n(\Omega_{n-1}), \quad\Omega_0\in R^d,\tag{1} \] \noindent where \(f_n\) are the maps, \(f_n:R^d\rightarrow R^d\) for stages \(n=1,2,\dots\). The goal is to construct supersets (enclosures) for the orbit of (1) such that the overestimation is kept small. (1) is not immediately suitable for the iteration on a computer. Therefore, the overestimation of the range \(f_n(\Omega_{n-1})\) at every stage \(n\) by wrapping it into a feasible superset is necessary to get a result suitable for the iteration on a computer (marching method). A collection of such sets and maps \(F_n\) operating on this collection are selected such that \(f_n(\Theta)\subseteq F_n(\Theta)\) for \(\Omega_0\subseteq\Theta_0\). Such maps are called extensions of \(f_n\) over the particular collection. The iterates are enclosures for (1). The paper presents a marching method (called cascade reduction) which beats the wrapping effect effectively and efficiently by constructing high order zonotope extensions of the orbit. A zonotope is the Minkowski sum of straight line segments. The number of line segments is always a multiple \(m\) of \(d\) here. The integer \(m\) is a freely selected measure for geometric complexity and is considered as a performance parameter for the method. The cascade reduction for finite precision, parallel computing and sparse maps are analyzed. Several examples are supplied.
reviewed by: Lubomír Bakule
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 93A99
Mathematics Subject Classification ID: 65Q05
zbMATH DE Number: 1199878
zbMATH Keywords: discrete dynamic system
zbMATH Keywords: wrapping effect
zbMATH Keywords: zonotopes
zbMATH Keywords: numerical examples
zbMATH Keywords: orbit
zbMATH Keywords: marching method
zbMATH Keywords: cascade reduction
zbMATH Keywords: Minkowski sum of straight line segments
zbMATH Keywords: geometric complexity
zbMATH Keywords: performance
zbMATH Keywords: parallel computing
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02684450
OpenAlex ID: W2051110650
description: scientific article; zbMATH DE number 1199878