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Quadratic systems with an integrable saddle: A complete classification in the coefficient space R^12
https://ddd.uab.cat/record/150528
A quadratic polynomial differential system can be identified with a single point of R12 through the coefficients. Using the algebraic invariant theory we classify all the quadratic polynomial differential systems of R12 having an integrable saddle. We show that there are only 47 topologically different phase portraits in the Poincar´e disc associated to this family of quadratic systems up to a reversal of the sense of their orbits. Moreover each one of these 47 representatives is determined by a set of affine invariant conditions. Artés, Joan CarlesFri, 06 May 2016 08:49:22 GMThttps://ddd.uab.cat/record/1505282012Liouvillian first integrals of quadratic-linear polynomial differential systems
https://ddd.uab.cat/record/150435
For a large class of quadratic–linear polynomial differential systems with a unique singular point at the origin having non-zero eigenvalues, we classify the ones which have a Liouvillian first integral, and we provide the explicit expression of them. Llibre, JaumeFri, 06 May 2016 08:30:49 GMThttps://ddd.uab.cat/record/1504352011Planar vector fields with a given set of orbits
https://ddd.uab.cat/record/150414
We determine all the C 1 planar vector fields with a given set of orbits of the form y − y(x) = 0 satisfying convenient assumptions. The case when these orbits are branches of an algebraic curve is also study. We show that if a quadratic vector field admits a unique irreducible invariant algebraic curve g(x, y) = S∑j=0 aj (x)y S−j = 0 with S branches with respect to the variable y, then the degree of the polynomial g is at most 4S. Llibre, JaumeFri, 06 May 2016 08:30:48 GMThttps://ddd.uab.cat/record/1504142011From topological to geometric equivalence in the classification of singularities at infinity for quadratic vector fields
https://ddd.uab.cat/record/145356
In the topological classification of phase portraits 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 distinction are however important in the production of limit cycles close to the foci in perturbations of the systems. The distinction between the one direction node and the two directions node, which plays a role in understanding the behavior of solution curves around the singularities at infinity, is also missing in the topological classification. In this work we introduce the notion of equivalence relation of singularities which incorporates these important purely algebraic features. The equivalence relation is finer than the one and also finer than the equivalence relation introduced in J_L. We also list all possibilities we have for singularities finite and infinite taking into consideration these finer distinctions and introduce notations for each one of them. Our long term goal is to use this finer equivalence relation to classify the quadratic family according to their different configurations of singularities, finite and infinite. In this work we accomplish a first step of this larger project. We give a complete global classification, using the equivalence relation, of the whole quadratic class according to the configuration of singularities at infinity of the systems. Our classification theorem is stated in terms of invariant polynomials and hence it can be applied to any family of quadratic systems with respect to any particular normal form. The theorem we give also contains the bifurcation diagram, done in the 12-parameter space, of the configurations of singularities at infinity, and this bifurcation set is algebraic in the parameter space. To determine the bifurcation diagram of configurations of singularities at infinity for any family of quadratic systems, given in any normal form, becomes thus a simple task using computer algebra calculations. Artés, Joan CarlesTue, 12 Jan 2016 16:38:11 GMThttps://ddd.uab.cat/record/1453562015Quadratic systems with invariant straight lines of total multiplicity two having Darboux invariants
https://ddd.uab.cat/record/145345
In this paper we present the global phase portraits in the Poincaré disc of the planar quadratic polynomial systems which admit invariant straight lines with total multiplicity two and Darboux invariants. Llibre, JaumeTue, 12 Jan 2016 16:38:10 GMThttps://ddd.uab.cat/record/1453452015Global configurations of singularities for quadratic differential systems with exactly three finite singularities of total multiplicity four
https://ddd.uab.cat/record/145286
In this article we obtain the geometric classification of singularities, finite and infinite, for the two subclasses of quadratic differential systems with total finite multiplicity m_f=4 possessing exactly three finite singularities, namely: systems with one double real and two complex simple singularities (31 configurations) and (ii) systems with one double real and two simple real singularities (265 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 classes of quadratic systems. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants. This gives an algorithm for determining the geometric configuration of singularities for any system in anyone of the two subclasses considered. Artés, Joan CarlesTue, 12 Jan 2016 16:38:06 GMThttps://ddd.uab.cat/record/1452862015Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields
https://ddd.uab.cat/record/118321
We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial vector fields. The concept of moduli space is discussed in the last section and we indicate its value in understanding the dynamics of families of such systems. Our interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated in the proof of our theorem in the section next to last. These concepts have proven their worth in a number of classification results, among them the most recent work on the geometric classification of the whole class of quadratic vector fields, according to their configurations of infinite singularities. An analog work including both finite and infinite singularities of the whole quadratic class, joint work with J. C. Artés, J. Llibre, and N. Vulpe, is in progress. Schlomiuk, DanaMon, 19 May 2014 16:31:48 GMThttps://ddd.uab.cat/record/1183212014