Global configurations of singularities for quadratic differential systems with total finite multiplicity three and at most two real singularities
| Autor: | Nicolae Vulpe, Jaume Llibre, Dana Schlomiuk, Joan Carles Artés |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
Poincaré compactification
Geometric equivalence relation Applied Mathematics Mathematical analysis Bifurcation diagram Configuration of singularities Affine invariant polynomials Singularity Quadratic equation Infinite and finite singularities Discrete Mathematics and Combinatorics Equivalence relation Gravitational singularity Invariant (mathematics) Algebraic number Quadratic differential Mathematics Quadratic vector fields |
| Zdroj: | Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona Recercat. Dipósit de la Recerca de Catalunya instname |
| Popis: | In this work we consider the problem of classifying all configurations of singularities, finite and infinite, of quadratic differential systems, with respect to the geometric equivalence relation defined in Artes et al. (Rocky Mount J Math, 2014). 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 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 us 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 [15]. 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 $$m_f$$ of finite singularities less than or equal to one. That work was continued in [4] where the geometric classification was done for the case $$m_f=2$$ . The case $$m_f=3$$ has been split in two separate papers because of its length. The subclass of three real distinct singular points was done in [5] and we complete this case here. In this article we obtain geometric classification of singularities, finite and infinite, for the remaining three subclasses of quadratic differential systems with $$m_f=3$$ namely: (i) systems with a triple singularity (19 configurations); (ii) systems with one double and one simple real singularities (62 configurations) and (iii) systems with one real and two complex singularities (75 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 invariant polynomials. This provides an algorithm for computing the geometric configuration of singularities for any quadratic system in this class. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|