Relationships between the discriminant curve and other bifurcation diagrams

L. P. Simon, Elizabeth Hild, Henrik Farkas

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

The parameter dependence of the number and type of the stationary points of an ODE is considered. The number of the stationary points is determined by the saddle-node (SN) bifurcation set and their type (e.g., stability) is given by another bifurcation diagram (e.g., Hopf bifurcation set). The relation between these bifurcation curves on the parmeter plane is investigated. It is shown that the 'cross-shaped diagram', when the Hopf bifurcation curve makes a loop around a cusp point of the SN curve, is typical in some sense. It is proved that the two bifurcation curves meet tangentially at their common points (Takens-Bogdanov point), and these common points persist as a third parameter is varied. An example is shown that exhibits two different types of 3-codimensional degenerate Takens-Bogdanov bifurcation.

Original languageEnglish
Pages (from-to)245-265
Number of pages21
JournalJournal of Mathematical Chemistry
Volume29
Issue number4
DOIs
Publication statusPublished - May 2001

Fingerprint

Hopf bifurcation
Bifurcation (mathematics)
Bifurcation Diagram
Discriminant
Bifurcation Curve
Bifurcation Set
Curve
Stationary point
Set theory
Hopf Bifurcation
Bogdanov-Takens Bifurcation
Saddle-node Bifurcation
Saddle
Cusp
Diagram
Relationships
Vertex of a graph

Keywords

  • Cross-shaped diagram
  • Hopf bifurcation
  • Parametric representation method
  • Singularity set
  • Takens-Bogdanov bifurcation

ASJC Scopus subject areas

  • Chemistry(all)
  • Applied Mathematics

Cite this

Relationships between the discriminant curve and other bifurcation diagrams. / Simon, L. P.; Hild, Elizabeth; Farkas, Henrik.

In: Journal of Mathematical Chemistry, Vol. 29, No. 4, 05.2001, p. 245-265.

Research output: Contribution to journalArticle

Simon, L. P. ; Hild, Elizabeth ; Farkas, Henrik. / Relationships between the discriminant curve and other bifurcation diagrams. In: Journal of Mathematical Chemistry. 2001 ; Vol. 29, No. 4. pp. 245-265.
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