Geometric configurations of singularities for quadratic differential systems with total finite multiplicity lower than 2
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 Sciences of Moldova. Institute of Mathematics and Computer Science)

Data: 2013
Resum: In [3] we classified globally the configurations of singularities at infinity of quadratic differential systems, with respect to the geometric equivalence relation. The global classification of configurations of finite singularities was done in [2] modulo the coarser topological equivalence relation for which no distinctions are made between a focus and a node and neither are they made between a strong and a weak focus or between foci of different orders. These 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 us to incorporates all these important purely algebraic features. This equivalence relation is also finer than the qualitative equivalence relation introduced in [19]. In this article we initiate the joint classification of configurations of singularities, finite and infinite, using the finer geometric equivalence relation, for the subclass of quadratic differential systems possessing finite singularities of total multiplicity mf ≤ 1. We obtain 84 geometrically distinct configurations of singularities for this family. We also give here the global bifurcation diagram, with respect to the geometric equivalence relation, of configurations of singularities, both finite and infinite, for this class of systems. This bifurcation set is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants. The results can therefore be applied for any family of quadratic systems, given in any normal form. Determining the configurations of singularities for any family of quadratic systems, becomes thus a simple task using computer algebra calculations.
Ajuts: European Commission 316338
Nota: Agraïments/Ajudes: 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.
Drets: Tots els drets reservats.
Llengua: Anglès
Document: Article ; recerca ; Versió acceptada per publicar
Matèria: Quadratic vector fields ; Infinite and finite singularities ; Affine invariant polynomials ; Poincaré compactification ; Configuration of singularities ; Geometric equivalence relation
Publicat a: Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica., Vol. 71 Núm. 1 (2013) , p. 72-124, ISSN 1024-7696

40 p, 1.5 MB

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