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Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities
Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Schlomiuk, Dana (Université de Montréal. Département de Mathématiques et de Statistiques)

Vulpe, Nicolae (Academy of Science of Moldova. Institute of Mathematics and Computer Science)

Date: | 2013 |

Abstract: | In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddles) of different orders. Such distinctions are, however, important in the production of limit cycles close to the foci (or loops) in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. The geomet ric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in a work published in 2013 when the classification was done for systems with total multiplicity m f of finite singularities less than or equal to one. That work was continued in an article which is due to appear in 2014 where the geometric classification of configurations of singularities was done for the case m f = 2. In this article we go one step further and obtain the geometric classification of singularities, finite and infinite, for the subclass mentioned above. We obtain 147 geometrically distinct configurations of singularities for this family. We give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, a fact which gives us an algorithm for determining the geometric configuration of singularities for any quadratic system in this particular class. |

Note: | Agraïments: The third author is supported by NSERC. The fourth author is also supported by the grant 12.839.08.05F from SCSTD of ASM and partially by NSERC. |

Note: | Número d'acord de subvenció MINECO/MTM2008-03437 |

Note: | Número d'acord de subvenció AGAUR/2009/SGR-410 |

Note: | Número d'acord de subvenció EC/FP7/2012/318999 |

Note: | Número d'acord de subvenció EC/FP7/2012/316338 |

Rights: | Tots els drets reservats. |

Language: | Anglès |

Document: | article ; recerca ; acceptedVersion |

Subject: | Geometric equivalence relation ; Quadratic vector fields ; Singularities |

Published in: | Journal of fixed point theory and applications, Vol. 14 Núm. 2 (2013) , p. 555-618, ISSN 1661-7746 |

DOI: 10.1007/s11784-014-0175-2

Postprint 43 p, 1.7 MB |

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Record created 2017-01-23, last modified 2020-11-01