### 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 language | English |
---|---|

Pages (from-to) | 245-265 |

Number of pages | 21 |

Journal | Journal of Mathematical Chemistry |

Volume | 29 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 2001 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Chemistry*,

*29*(4), 245-265. https://doi.org/10.1023/A:1010943118331

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

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, vol. 29, no. 4, pp. 245-265. https://doi.org/10.1023/A:1010943118331

}

TY - JOUR

T1 - Relationships between the discriminant curve and other bifurcation diagrams

AU - Simon, L. P.

AU - Hild, Elizabeth

AU - Farkas, Henrik

PY - 2001/5

Y1 - 2001/5

N2 - 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.

AB - 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.

KW - Cross-shaped diagram

KW - Hopf bifurcation

KW - Parametric representation method

KW - Singularity set

KW - Takens-Bogdanov bifurcation

UR - http://www.scopus.com/inward/record.url?scp=0035539375&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035539375&partnerID=8YFLogxK

U2 - 10.1023/A:1010943118331

DO - 10.1023/A:1010943118331

M3 - Article

AN - SCOPUS:0035539375

VL - 29

SP - 245

EP - 265

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

IS - 4

ER -