Use of the parametric representation method in revealing the root structure and hopf bifurcation

Henrik Farkas, L. P. Simon

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

In practical applications of dynamical systems, it is often necessary to determine the number and the stability of the stationary states. The parameric respresentation method is a useful tool in such problems. Consider the two parameter families of functions:f(x) =uo +u1x +g(x), where uo and u1 are the parameters. We are interested in the number of zeros as well as in the stability. We want to determine the "stable region" on the parameter plane, where the real parts of the roots of f are negative. The D-curve (along which the discriminant of f is zero) helps us. We applied the method to the cases of cubic and quartic equation, giving pictorial meaning to the root structure. In this respect, the R-curves and the I-curves (along which the sum or difference, respectively, of two zeros is constant) also have a significance. Using these concepts, we established a relation between the (n - 1)th Routh-Hurwitz condition and the Hopf bifurcation.

Original languageEnglish
Pages (from-to)323-339
Number of pages17
JournalJournal of Mathematical Chemistry
Volume9
Issue number4
DOIs
Publication statusPublished - Dec 1992

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Parametric Representation
Hopf bifurcation
Hopf Bifurcation
Roots
Curve
Zero
Dynamical systems
Stationary States
Quartic
Discriminant
Two Parameters
Dynamical system
Necessary

ASJC Scopus subject areas

  • Applied Mathematics
  • Chemistry(all)

Cite this

Use of the parametric representation method in revealing the root structure and hopf bifurcation. / Farkas, Henrik; Simon, L. P.

In: Journal of Mathematical Chemistry, Vol. 9, No. 4, 12.1992, p. 323-339.

Research output: Contribution to journalArticle

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