Joan Carles Artés

THE GEOMETRY OF QUADRATIC DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS OF SECOND ORDER

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilberts 16th problem [Hilbert, 1900, 1902], are still open for this class. In this article we make an interdisciplinary global study of the subclass which is the closure within real quadratic differential systems, of the family QW2 of all such systems which have a weak focus of second order. This class also includes the family of all quadratic differential systems possessing a weak focus of third order and topological equivalents of all quadratic systems with a center. The bifurcation diagram for this class, done in the adequate parameter space which is the three-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for , 95 for QW2, 20 having limit cycles but only three with the maximum number of limit cycles (two) within this class. The phase portraits are always represented in the Poincare disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of points which we suspect to be analytic corresponding to global separatrices which have connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. The global geometry of this class reveals interesting bifurcations phenomena; for example, all phase portraits with limit cycles in this class can be produced by perturbations of symmetric (reversible) quadratic systems with a center. Many other nonlinear phenomena displayed here form material for further studies.

Original article: Piecewise linear differential systems with two real saddles

In this paper we study piecewise linear differential systems formed by two regions separated by a straight line so that each system has a real saddle point in its region of definition. If both saddles are conveniently situated, they produce a transition flow from a segment of the splitting line to another segment of the same line, and this produces a generalized singular point on the line. This point is a focus or a center and there can be found limit cycles around it. We are going to show that the maximum number of limit cycles that can bifurcate from this focus is two. One of them appears through a Hopf bifurcation and the second when the focus becomes a node by means of the sliding.

SINGULAR POINTS OF QUADRATIC SYSTEMS: A COMPLETE CLASSIFICATION IN THE COEFFICIENT SPACE ℝ12

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is sti...

Quadratic vector fields with a weak focus of third order

We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle.

From topological to geometric equivalence in the classification of singularities at infinity for quadratic vector fields

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.

THE GEOMETRY OF QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS AND AN INVARIANT STRAIGHT LINE

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than a thousand papers were written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilberts 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we conduct a global study of the class QWI of all real quadratic differential systems which have a weak focus and invariant straight lines of total multiplicity of at least two. This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real projective space of the parameters of this form. The bifurcation diagram yields 73 phase portraits for systems in QWI plus 26 additional phase portraits with the center at its border points. Algebraic invariants are used to construct the bifurcation set. We show that all systems in QWI necessarily have their weak focus of order one and invariant straight lines of total multiplicity exactly two. The phase portraits are represented on the Poincare disk. The bifurcation set is algebraic and all points in this set are points of bifurcation of singularities. We prove that there is no phase portrait with limit cycles in this class but that there is a total of five phase portraits with graphics, four having the invariant line as a regular orbit and one phase portrait with an infinity of graphics which are all homoclinic loops inside a heteroclinic graphic with two singularities, both at infinity.

GLOBAL PHASE PORTRAITS OF QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS WITH A SEMI-ELEMENTAL TRIPLE NODE

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, Hilberts 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family of all real quadratic polynomial differential systems which have a semi-elemental triple node (triple node with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this form. This bifurcation diagram yields 28 phase portraits for systems in counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincare disk. The bifurcation set is not only algebraic due to the presence of a surface found numerically. All points in this surface correspond to connections of separatrices.

THE GEOMETRY OF QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS WITH A FINITE AND AN INFINITE SADDLE-NODE (A,B)

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, Hilberts 16th problem [Hilbert, 1900, 1902], are still open for this family. Our goal is to make a global study of the family QsnSN of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node 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 in the origin of the plane with the eigenvectors on the axes and with the eigenvector associated with the zero eigenvalue on the horizontal axis 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 and we give the bifurcation diagram of their closure with respect to specific normal forms, in the three-dimensional real projective space. The subfamilies (A) and (B) have already been studied [Artes et al., 2013b] and in this paper we provide the complete study of the geometry of the last family (C). The bifurcation diagram for the subfamily (C) yields 371 topologically distinct phase portraits with and without limit cycles for systems in the closure within the representatives of QsnSN(C) given by a chosen normal form. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincare disk. The bifurcation set of is not only algebraic due to the presence of some surfaces found numerically. All points in these surfaces correspond to either connections of separatrices, or the presence of a double limit cycle.

Uniform isochronous cubic and quartic centers: Revisited

Abstract In this paper we completed the classification of the phase portraits in the Poincare disc of uniform isochronous cubic and quartic centers previously studied by several authors. There are three and fourteen different topological phase portraits for the uniform isochronous cubic and quartic centers respectively.

Proof of Theorem 1.1(b)

We have already developed all the needed tools for doing the classification of topologically possible structurally unstable quadratic systems of codimension one∗. Along with the classification, we will discard many systems that are topologically possible; we can show their impossibility by means of their unfoldings or other criteria already described. Some phase portraits will pass these main filters and will appear as possible. However, we will discard some of them later on in Chap. 6 using more specific lemmas for each of them. We have preferred not to include these lemmas in this chapter or in Chap. 3 in order not to disturb the flow of this classification which is already quite long and tedious even this will force a renumbering of the cases.