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Invariant conditions for phase portraits of quadratic systems with complex conjugate invariant lines meeting at a finite point
https://ddd.uab.cat/record/232170
The goal of this article is to give invariant necessary and sufficient conditions for a quadratic system, presented in whatever normal form, to have anyone of 17 out of the 20 phase portraits of the family of quadratic systems with two complex conjugate invariant lines intersecting at a finite real point. The systems in this family have a maximum of one limit cycle. Among the 17 phase portraits we have two with limit cycles. We also give invariant necessary and sufficient conditions for a system to have one of the three remaining phase portraits, out of which one has a limit cycle and another one a homoclinic loop. In the region R determined by these last conditions, due to the presence of systems with a homoclinic loop, an analytic condition, the three phase portraits cannot be separated by algebraic conditions in terms of invariant polynomials. We also give the bifurcation diagram of this family, outside the region R, in the twelve parameter space of coefficients of the systems. Artés, Joan CarlesMon, 14 Sep 2020 15:33:32 GMThttps://ddd.uab.cat/record/2321702020Geometric and algebraic classification of quadratic differential systems with invariant hyperbolas
https://ddd.uab.cat/record/222603
Let QSH be the whole class of non-degenerate planar quadratic diﬀerential systems possessing at least one invariant hyperbola. We classify this family of systems, modulo the action of the group of real aﬃne transformations and time rescaling, according to their geometric properties encoded in the conﬁgurations of invariant hyperbolas and invariant straight lines which these systems possess. The classiﬁcation is given both in terms of algebraic geometric invariants and also in terms of aﬃne invariant polynomials and it yields a total of 205 distinct such conﬁgurations. We have 162 conﬁgurations for the subclass QSH(η>0) of systems which possess three distinct real singularities at inﬁnity, and 43 conﬁgurations for the subclass QSH(η=0) of systems which possess either exactly two distinct real singularities at inﬁnity or the line at inﬁnity ﬁlled up with singularities. The algebraic classiﬁcation, based on the invariant polynomials, is also an algorithm which makes it possible to verify for any given real quadratic diﬀerential system if it has invariant hyperbolas or not and to specify its conﬁguration of invariant hyperbolas and straight lines. Oliveira, Regilene D. S.Tue, 26 May 2020 14:58:04 GMThttps://ddd.uab.cat/record/2226032017Global topological configurations of singularities for the whole family of quadratic differential systems
https://ddd.uab.cat/record/221281
In Artés et al. (Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case. Birkhäuser, Basel, 2019) the authors proved that there are 1765 different global geometrical configurations of singularities of quadratic differential systems in the plane. There are other 8 configurations conjectured impossible, all of them related with a single configuration of finite singularities. This classification is completely algebraic and done in terms of invariant polynomials and it is finer than the classification of quadratic systems according to the topological classification of the global configurations of singularities, the goal of this article. The long term project is the classification of phase portraits of all quadratic systems under topological equivalence. A first step in this direction is to obtain the classification of quadratic systems under topological equivalence of local phase portraits around singularities. In this paper we extract the local topological information around all singularities from the 1765 geometric equivalence classes. We prove that there are exactly 208 topologically distinct global topological configurations of singularities for the whole quadratic class. The 8 global geometrical configurations conjectured impossible do not affect this number of 208. From here the next goal would be to obtain a bound for the number of possible different phase portraits, modulo limit cycles. Artés, Joan CarlesWed, 15 Apr 2020 15:23:02 GMThttps://ddd.uab.cat/record/2212812020Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities
https://ddd.uab.cat/record/169420
In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer 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 (or loops) 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. The geomet ric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in a work published in 2013 when 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 an article which is due to appear in 2014 where the geometric classification of configurations of singularities was done for the case m f = 2. In this article we go one step further and obtain the geometric classification of singularities, finite and infinite, for the subclass mentioned above. We obtain 147 geometrically distinct configurations of singularities for this family. We give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class 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, a fact which gives us an algorithm for determining the geometric configuration of singularities for any quadratic system in this particular class. Artés, Joan CarlesMon, 23 Jan 2017 15:21:44 GMThttps://ddd.uab.cat/record/1694202013Geometric configurations of singularities for quadratic differential systems with total finite multiplicity m_f=2
https://ddd.uab.cat/record/150698
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 [3]. This relation is finer 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 to incorporates all these important geometric features which can be expressed in purely algebraic terms. This equivalence relation is also finer than the qualitative equivalence relation introduced in [17]. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in [4] where the classification was done for systems with total multiplicity mf of finite singularities less than or equal to one. In this article we continue the work initiated in [4] and obtain the geometric classification of singularities, finite and infinite, for the subclass of quadratic differential systems possessing finite singularities of total multiplicity mf = 2. We obtain 197 geometrically distinct configurations of singularities for this family. We also give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class 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. The results can therefore be applied for any family of quadratic systems in this class, given in any normal form. Determining the geometric configurations of singularities for any such family, becomes thus a simple task using computer algebra calculations. Artés, Joan CarlesFri, 06 May 2016 08:59:50 GMThttps://ddd.uab.cat/record/1506982014Global configurations of singularities for quadratic differential systems with total finite multiplicity three and at most two real singularities
https://ddd.uab.cat/record/150677
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 [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 (or loops) 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 [19]. 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. The case mf = 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 mf = 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 (74 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. Artés, Joan CarlesFri, 06 May 2016 08:59:49 GMThttps://ddd.uab.cat/record/1506772014Global configurations of singularities for quadratic differential systems with exactly two finite singularities of total multiplicity four
https://ddd.uab.cat/record/150673
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. Artés, Joan CarlesFri, 06 May 2016 08:59:48 GMThttps://ddd.uab.cat/record/1506732014Geometric configurations of singularities for quadratic differential systems with total finite multiplicity lower than 2
https://ddd.uab.cat/record/150578
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. Artés, Joan CarlesFri, 06 May 2016 08:54:14 GMThttps://ddd.uab.cat/record/1505782013On the limit cycles bifurcating from an ellipse of a quadratic center
https://ddd.uab.cat/record/145371
Consider the class of all quadratic centers whose period annulus has a periodic solution whose phase curve is an ellipse E. The period annulus of any of such quadratic centers has cyclicity at least one, and this one is due to a family of algebraic limit cycles(formed by ellipses) bifurcating from the ellipse E. quadratic systems, quadratic vector ﬁelds, quadratic center, periodic orbit, limit cycle, bifurcation from center, cyclicity of the period annulus, inverse integrating factor. Llibre, JaumeTue, 12 Jan 2016 16:38:12 GMThttps://ddd.uab.cat/record/1453712015From 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/1453562015Global 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