Global configurations of singularities for quadratic differential systems with exactly two finite singularities of total multiplicity four
| Autor: | Jaume Llibre, Joan Carles Artés, Dana Schlomiuk, Alex C. Rezende, Nicolae Vulpe |
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| Rok vydání: | 2021 |
| Předmět: |
Poincaré compactification
Geometric equivalence relation Applied Mathematics Mathematical analysis quadratic vector fields Multiplicity (mathematics) infinite and finite singularities geometric equivalence relation Bifurcation diagram Configuration of singularities Affine invariant polynomials Quadratic equation Infinite and finite singularities configuration of singularities QA1-939 Equivalence relation Gravitational singularity Algebraic number Quadratic differential poincaré compactification Bifurcation Mathematics affine invariant polynomials Quadratic vector fields |
| Zdroj: | Recercat. Dipósit de la Recerca de Catalunya instname Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) Electronic Journal of Qualitative Theory of Differential Equations, Vol 2014, Iss 60, Pp 1-43 (2014) |
| Popis: | 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 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.. 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. |
| Databáze: | OpenAIRE |
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