### 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) =u_{o} +u_{1}x +g(x), where u_{o} and u_{1} 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 language | English |
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Pages (from-to) | 323-339 |

Number of pages | 17 |

Journal | Journal of Mathematical Chemistry |

Volume | 9 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1992 |

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### 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.

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, vol. 9, no. 4, pp. 323-339. https://doi.org/10.1007/BF01166096

}

TY - JOUR

T1 - Use of the parametric representation method in revealing the root structure and hopf bifurcation

AU - Farkas, Henrik

AU - Simon, L. P.

PY - 1992/12

Y1 - 1992/12

N2 - 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.

AB - 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.

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U2 - 10.1007/BF01166096

DO - 10.1007/BF01166096

M3 - Article

VL - 9

SP - 323

EP - 339

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

IS - 4

ER -