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| Home > Articles > Published articles > Global configurations of singularities for quadratic differential systems with exactly two finite singularities of total multiplicity four |
(Universitat Autònoma de Barcelona. Departament de Matemàtiques) (Universitat Autònoma de Barcelona. Departament de Matemàtiques) (Universidade de São Paulo)| Additional title: | Geometric classification of configurations of singularities with total finite multiplicity four for a class of quadratic systems |
| Date: | 2014 |
| Abstract: | In this work we consider the problem of classifying all configurations of singularities, both finite and infinite of quadratic differential systems, with respect to the geometric equivalence relation defined in [2]. This relation is deeper 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 in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows to incorporate all these important geometric features which can be expressed in purely algebraic terms. This equivalence relation is also deeper than the qualitative equivalence relation introduced in [20]. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in [3] where the classification was done for systems with total multiplicity mf of finite singularities less than or equal to one. That work was continued in [4] where the geometric classification was done for the case mf = 2 and two more papers [5] and [6], which cover the case mf = 3. In this article we obtain the geometric classification of singularities, finite and infinite, for the three subclasses of quadratic differential systems with mf = 4 possessing exactly two finite singularities, namely: (i) systems with two double complex singularities (18 configurations); (ii) systems with two double real singularities (33 configurations) and (iii) systems with one triple and one simple real singularities (123 configurations). We also give here the global bifurcation diagrams of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for these subclasses 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, fact which gives an algorithm for determining the geometric configuration of singularities for any quadratic system. |
| Grants: | European Commission 318999 European Commission 316338 Ministerio de Economía y Competitividad MTM2008-03437 Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2013/SGR-568 |
| Note: | El títol de la versió pre-print de l'article és: Geometric classification of configurations of singularities with total finite multiplicity four for a class of quadratic systems |
| Note: | Agraïments/Ajudes: The third author is supported by CAPES/DGU BEX 9439-12-9. The fourth and fifth author are supported by NSERC-RGPIN (8528-2010). The fifth author is also supported by the grant 12.839.08.05F from SCSTD of ASM.. |
| Rights: | Tots els drets reservats.  |
| Language: | Anglès |
| Document: | Article ; recerca ; Versió acceptada per publicar |
| Subject: | Affine invariant polynomials ; Configuration of singularities ; Geometric equivalence relation ; Infinite and finite singularities ; Poincaré compactification ; Quadratic vector fields |
| Published in: | Electronic Journal of Qualitative Theory of Differential Equations, Vol. 60 (2014) , p. 1-43, ISSN 1417-3875 |
36 p, 1.5 MB |
