On the preservation of properties of differential equations under discretization
type: Item
instance of: scholarly article
title: On the preservation of properties of differential equations under discretization
zbMATH Open document ID: 0885.65089
author: L. Z. Fishman
published in: Doklady Mathematics
publication date: 1998-04-20T00:00:00Z
review text: Consider the system \[ \dot x=F(x,\alpha), \tag{1} \] where \(x\) is an \(n\)-dimensional vector, \(n\geq 1\), \(F(x,\alpha)\) is a sufficiently smooth or analytic vector-function of \((x,\alpha)\), \(F(0,0)=0\), and \(\alpha\) is a parameter. Let, under discretization, system (1) be replaced by the discrete systems \[ \begin{aligned} \overline{x}&= x+h \sum_{i=1}^s b_iF(Y_i,\alpha), \tag{2}\\ \overline{x}&= x+hF (\overline{x},\alpha), \tag{3}\end{aligned} \] which are constructed by the \(s\)-stage \(p\)th-order Runge-Kutta method and by the implicit Euler method, where \[ Y_i=x+h \sum_{j=1}^s a_{ij} F(Y_j,\alpha), \qquad 1\leq i\leq s, \] \(a_{ij}\) and \(b_j\) are coefficients of the Runge-Kutta method, and \[ \sum_{i=1}^sb_i=1, \qquad s\geq p\geq 1,\quad h>0. \] We consider the problem of the preservation of the properties of equilibrium states of system (1), when it is replaced by the systems (2) and (3), if equilibrium states of system (1) have one zero root. We give sufficient conditions for the preservation of multiplicity, stability and bifurcations of equilibrium states of system (1) in the case \(n=1,2\) and \(n\geq 3\) for \(h\) being not necessarily sufficiently small. We present an example.
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34D30
zbMATH DE Number: 1099463
zbMATH Keywords: system
zbMATH Keywords: Runge-Kutta method
zbMATH Keywords: implicit Euler method
zbMATH Keywords: equilibrium states
zbMATH Keywords: stability
zbMATH Keywords: bifurcations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1099463