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The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (A,B)
https://ddd.uab.cat/record/150702
Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, Hilbert, 1902], are still open for this family. Our aim is to make a global study of the family QsnSN which is the closure within real quadratic differential systems of the family QsnSN of all such systems which have two semi-elemental saddle-nodes, one finite and one infinite formed by the collision of two infinite singular points. This family can be divided into three different subfamilies, all of them with the finite saddle-node at the origin of the plane with the eigenvectors on the axes and (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and time homotheties are three-dimensional (the closure is four-dimensional) and we give their bifurcation diagram with respect to a normal form. In this paper we provide the complete study of the geometry of the first two families, (A) and (B). The bifurcation diagram for the subfamily (A) yields 38 phase portraits for systems in QsnSN(A) (29 in QsnSN(A)) out of which only 3 have limit cycles and 13 possess graphics. The bifurcation diagram for the subfamily (B) yields 25 phase portraits for systems in QsnSN(B) (16 in QsnSN(B)) out of which 11 possess graphics. None of the 25 portraits has limit cycles. Case (C) will yield many more phase portraits and will be written separately in a forthcoming new paper. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of QsnSN(A) is formed by algebraic surfaces and one surface whose presence was detected numerically. All points in this surface correspond to connections of separatrices. The bifurcation set of QsnSN(B) is formed only by algebraic surfaces. Artés, Joan CarlesFri, 06 May 2016 08:59:50 GMThttps://ddd.uab.cat/record/1507022014Piecewise bounded quadratic systems in the plane
https://ddd.uab.cat/record/150697
In this paper we study the sliding mode of piecewise bounded quadratic systems in the plane given by a non-smooth vector field Z = (X, Y ). Analyzing the singular, crossing and sliding sets, we get the conditions which ensure that any solution, including the sliding one, is bounded. Llibre, JaumeFri, 06 May 2016 08:59:50 GMThttps://ddd.uab.cat/record/1506972014Global phase portraits of a SIS model
https://ddd.uab.cat/record/150612
In the qualitative theory of ordinary differential equations, we can find many papers whose objective is the classification of all the possible topological phase portraits of a given family of differential system. Most of the studies rely on systems with real parameters and the study consists of outlining their phase portraits by finding out some conditions on the parameters. Here, we studied a susceptible-infected-susceptible (SIS) model described by the differential system x˙ = −bxy − mx + cy + mk, y˙ = bxy − (m + c)y, where b, c, k, m are real parameters with b 6= 0, m 6= 0 [3]. Such system describes an infectious disease from which infected people recover with immunity against reinfection. The integrability of such system has already been studied by Nucci and Leach [8] and Llibre and Valls [6]. We found out two different topological classes of phase portraits. Oliveira, Regilene D. S.Fri, 06 May 2016 08:54:15 GMThttps://ddd.uab.cat/record/1506122013On the global flow of a 3--dimensional Lotka--Volterra system
https://ddd.uab.cat/record/150567
In the study of the black holes with Higgs field appears in a natural way the Lotka-Volterra differential system x˙= x(y − 1), y˙= y(1 + y − 2x2 − z2), z˙= zy, in R3. Here we provide the qualitative analysis of the flow of this system describing the α-limit set and the ω-limit set of all orbits of this system in the whole Poincar'e ball, i. e. we identify R3 with the interior of the unit ball of R3 centered at the origin and we extend analytically this flow to its boundary, i. e. to the infinity. Alavez-Ramírez, JustinoFri, 06 May 2016 08:49:24 GMThttps://ddd.uab.cat/record/1505672012Bifurcation values for a familiy of planar vector fields of degree five
https://ddd.uab.cat/record/145366
We study the number of limit cycles and the bifurcation diagram in the Poincar' sphere of a one-parameter family of planar diﬀerential equations of degree e ˙ ﬁve x = Xb (x) which has been already considered in previous papers. We prove that there is a value b∗ > 0 such that the limit cycle exists only when b ∈ (0, b∗ ) and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length 27/1000 where b∗ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic curves without contact for the ﬂow of the diﬀerential equation. These curves are obtained using analytic information about the separatrices of the inﬁnite critical points of the vector ﬁeld. To prove that the Bendixson-Dulac Theorem works we develop a method for studying whether one-parameter families of polynomials in two variables do not vanish based on the computation of the so called double discriminant. García Saldaña, Johanna DeniseTue, 12 Jan 2016 16:38:12 GMThttps://ddd.uab.cat/record/1453662015Quadratic 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/1453452015Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields
https://ddd.uab.cat/record/145320
As a natural continuation of the work done in [7] we provide the bifurcation diagrams for the global phase portraits in the Poincaré disk of all the Hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields. Colak, IlkerTue, 12 Jan 2016 16:38:09 GMThttps://ddd.uab.cat/record/1453202015Phase portraits of uniform isochronous quartic centers
https://ddd.uab.cat/record/145315
In this paper we classify the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin such that their nonlinear part is not homogeneous. Itikawa, JacksonTue, 12 Jan 2016 16:38:08 GMThttps://ddd.uab.cat/record/1453152015Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields
https://ddd.uab.cat/record/145294
This article deals with the bifurcation of polycycles and limit cycles within the 1-parameter families of planar vector fields X_m^k, defined by =y^3-x^2k 1,=-x my^4k 1, where m is a real parameter and k1 integer. The bifurcation diagram for the separatrix skeleton of X_m^k in function of m is determined and the one for the global phase portraits of (X^1_m)_mR is completed. Furthermore for arbitrary k1 some bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of X_m^k is found to be uniformly bounded independent of mR and the Hilbert number for (X_m^k)_mR, that thus is finite, is found to be at least one. Caubergh, MagdalenaTue, 12 Jan 2016 16:38:07 GMThttps://ddd.uab.cat/record/1452942015Topological 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