Numerical dynamics of a nonstandard finite difference method for a class of delay differential equations
type: Item
instance of: scholarly article
title: Numerical dynamics of a nonstandard finite difference method for a class of delay differential equations
zbMATH Open document ID: 1266.65116
author: Huan Su
author: Wenxue Li
author: Xiaohua Ding
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-02-21T00:00:00Z
review text: A nonstandard finite difference (NSFD) method for a class of delay differential equations is proposed. The method has some remarkable properties for any step size \(h=1/m\) such as: preservation of the qualitative behavior that includes the local stability of the equilibrium, the presence of Hopf bifurcations and their directions, and the stability of the closed invariant curves representing periodic solutions generated by bifurcation. The NSFD method applied to the delay differential equation provides a numerical discrete equation. The existence and the local stability of the Neimark-Sacker bifurcation at the fixed point for the numerical discrete equation is analysed. Moreover, an explicit algorithm for determining the direction of the Neimark-Sacker bifurcation is presented. In order to illustrate the effectiveness of the NSFD method, some numerical tests for a model of the survival of red blood cells in an animal are given, including a comparison with the Euler implicit method.
reviewed by: Alexandru Mihai Bica
Mathematics Subject Classification ID: 65L03
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K18
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34K13
zbMATH DE Number: 6138101
zbMATH Keywords: nonstandard finite difference methods
zbMATH Keywords: local stability
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: Neimark-Sacker bifurcation
zbMATH Keywords: numerical examples
zbMATH Keywords: delay differential equations
zbMATH Keywords: invariant curves
zbMATH Keywords: periodic solutions
zbMATH Keywords: algorithm
zbMATH Keywords: red blood cells
zbMATH Keywords: Euler implicit method
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.11.033
OpenAlex ID: W2077907080
DOI: 10.1016/J.JMAA.2012.11.033
description: scientific article; zbMATH DE number 6138101