Deciding Hopf bifurcations by quantifier elimination in a software-component architecture
type: Item
instance of: scholarly article
title: Deciding Hopf bifurcations by quantifier elimination in a software-component architecture
zbMATH Open document ID: 0965.65137
author: M'hammed El Kahoui
author: Andreas Weber
published in: Journal of Symbolic Computation
publication date: 2001-02-06T00:00:00Z
review text: The paper investigates the phenomenon of Hopf bifurcations for autonomous, deterministic ordinary differential equations with parameterized polynomial vector fields. The authors make use of ``quantifier elimination'' -- a symbolic method which has been developed and implemented by the second author, avoiding Lie-symmetry methods like in the book of \textit{P. J. Olver} [Applications of Lie groups to differential equations. Paperback ed. (English) Graduate Texts in Mathematics. 107. New York, NY: Springer. xxviii, 513 p. (2000; Zbl 0937.58026)] or differential Galois theory like that of \textit{M. F. Singer} [Computer algebra and differential equations, Colloq., Comput. Math. Appl., 3-57 (1988; Zbl 0713.12005)] as well. The used techniques rely on Hurwitz determinants, applying the theory of sub-resultant sequences, of Groebner bases and quantifier elimination algorithms. A series of examples from existing literature illustrates the use of Java-based architectures to evaluate the related algebraic conditions on Hopf bifurcations. It is worth noting that the technique of quantifier elimination on real closed fields is due to \textit{A. Tarski} [A decision method for elementary algebra and geometry. 2nd ed. (English) Berkeley. University of California Press. III, 63 p. (1951, Zbl 0044.25102)].
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 65P30
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 37M20
zbMATH DE Number: 1513056
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: ODEs with polynomial vector fields
zbMATH Keywords: quantifier elimination
zbMATH Keywords: symbolic methods
zbMATH Keywords: symmetry methods
zbMATH Keywords: Groebner basis methods
zbMATH Keywords: Hurwitz determinants criterion
zbMATH Keywords: Java-based software architecture
describes a project that uses: QEPCAD
describes a project that uses: SYMMGRP
describes a project that uses: REDLOG
MaRDI profile type: Publication
full work available at URL: jsco.1999.0353
OpenAlex ID: W2119763521
Wikidata QID: Q29012983
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DOI: 10.1006/JSCO.1999.0353
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description: scientific article; zbMATH DE number 1513056