Dana Schlomiuk

Algebraic particular integrals, integrability and the problem of the center

In this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields. We first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center: these curves control the changes in the systems as parameters vary. The bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered, namely the real four-dimensional projective space. Next, we consider the known four algebraic conditions for the center for quadratic vector fields. One of them says that the system is Hamiltonian, a condition which has a clear geometric meaning. We determine the geometric meaning of the remaining other three algebraic conditions (I), (II), (III)

Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields

Due to the algebraic form of polynomial vector fields, questions and techniques of an algebraic or algebro-geometric nature are suitable for this setting. In these notes we discuss some of them together with applications.

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.

Planar quadratic vector fields with invariant lines of total multiplicity at least five

In this article we consider the action of the real affine group and time rescaling on real planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of real quadratic differential systems with at least five invariant lines. These conditions are such that no matter how a system may be presented, one can verify by using them whether the system has or does not have at least five invariant lines and to check to which orbit (or family of orbits) it belongs.

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

In this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert’s 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles. Received by the editors October 12, 2001; revised March 13, 2003. The first author is partially supported by a MCYT grant number BFM 2002-04236-C02-02 and by a CICYT grant number 2001SGR 003173, and the second is partially supported by NSERC and by Quebec Education Ministry, and was also supported by a grant from the Ministerio de Educación y Cultura of Spain. AMS subject classification: Primary: 34C40, 51F14; secondary: 14D05, 14D25. c ©Canadian Mathematical Society 2004.

Integrals and Phase Portraits of Planar Quadratic Differential Systems With Invariant Lines of at Least Five Total Multiplicity

In this article we prove that all real quadratic differential systems dx dt = p(x, y), dy dt = q(x, y) with gcd(p, q) = 1, having invariant lines of total multiplicity at least five and a finite set of singularities at infinity, are Darboux integrable having integrating factors whose inverses are polynomials over R. We also classify these systems under two equivalence relations: 1) topological equivalence and 2) equivalence of their associated cubic projective differential equations when cubic projective differential equations are acted upon by the group PGL(2,R). For each one of the 28 topological classes obtained we give necessary and sufficient conditions for such a system with invariant lines to belong to this class, in terms of its coefficients in R12. Résumé Dans cet article nous prouvons que tous les systèmes différentiels réels dx dt = p(x, y), dy dt = q(x, y), gcd(p, q) = 1, ayant des droites invariantes de multiplicité totale au moins cinq, sont Darboux intégrable possédant des facteurs intégrant dont les inverses sont des polynomes sur R. Nous classifions ces systèmes par rapport à deux relations d’équivalence : 1) l’équivalence topologique et 2) l’équivalence de leurs équations différentielles cubiques projectives sous l’action du groupe PGL(2,R). Pour chaqune des 28 classes topologiques obtenues nous donnons des conditions nécessaires et suffisantes pour qu’un tel système ayant des droites invariantes appartienne à cette classe, en terms de ses coefficients dans R12.

Summing up the dynamics of quadratic Hamiltonian systems with a center

In this work we study the global geometry of planar quadratic Hamiltonian systems with a center and we sum up the dynamics of these systems in geometrical terms. For this we use the algebro-geometric concept of multiplicity of intersection Ip(POQ) of two complex projective curves P(xO yO z) = 0, Q(xO yO z) = 0 at a point p of the plane. This is a convenient concept when studying polynomial systems and it could be applied for the analysis of other classes of nonlinear systems. The work of the second author was partially supported by NSERC and both authors were partially supported by Quebec Education Ministry. Received by the editors March 16, 1995. AMS subject classification: 34C, 58F. c Canadian Mathematical Society 1997. 582

On the Geometric Structure of the Class of Planar Quadratic Differential Systems

In this work we are interested in the global theory of planar quadratic differential systems and more precisely in the geometry of this whole class. We want to clarify some results and methods such as the isocline method or the role of rotation parameters. To this end, we recall how to associate a pencil of isoclines to each quadratic differential equation. We discuss the parameterization of the space of regular pencils of isoclines by the space of its multiple base points and the equivariant action of the affine group on the fibration of the space of regular quadratic differential equations over the space of regular pencils of isoclines. This fibration is principal, with a projective group as structural group, and we prove that there exits an open cone in its Lie algebra whose elements generate rotation parameter families. Finally we use this geometric approach to construct specific families of quadratic differential equations depending in a nonlinear way of parameters which have a geometric meaning: they parameterize the set of singular points or are rotation parameters leaving fixed this set.

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.

Basic algebro-geometric concepts in the study of planar polynomial vector fields

In this work we show that basic algebro-geometric concepts such as the concept of intersection multiplicity of projective curves at a point in the complex projective plane, are needed in the study of planar polynomial vector fields and in particular in summing up the information supplied by bifurcation diagrams of global families of polynomial systems. Algebro-geometric concepts are helpful in organizing and unifying in more intrinsic ways this information.