Planar Quadratic Vector Fields Research Articles

Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves

Abstract The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R 2 {{\mathbb{R}}}^{2} , particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.

Hilbert’s 16th problem on a period annulus and Nash space of arcs

AbstractThis paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisorEof the canonical blow-upBIℂnof the Bautin idealI. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcsArc(BIℂn,E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

Quadratic systems with an invariant conic having Darboux invariants

The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real affine transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these normal forms, we complete the characterization of the phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic [Formula: see text]: [Formula: see text], and a Darboux invariant of the form [Formula: see text] with [Formula: see text].

Non-smooth quadratic centers defined in two arbitrary sectors

In this paper we analyze the center–focus problem of some families of piecewise planar quadratic vector fields on two zones of R2. The zones we consider are two unbounded sectors defined by an arbitrary angle α and a fixed vertex. We also assume that each vector field share a common weak focus singularity at the vertex of the boundary. We observe how the center variety depends on the angle α.

Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields

In this article, we consider the class [Formula: see text] of all real quadratic differential systems [Formula: see text], [Formula: see text] with gcd (p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class [Formula: see text] so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.

Finite Cyclicity of Some Graphics Through a Nilpotent Point of Saddle Type Inside Quadratic Systems

In this paper we show the finite cyclicity of the two graphics $$(I_{12}^1)$$ and $$(I_{13}^1)$$ through a triple nilpotent point of saddle type inside quadratic vector fields. These results contribute to the program launched in 1994 by Dumortier, Roussarie and Rousseau (DRR program) to show the existence of a uniform upper bound for the number of limit cycles for planar quadratic vector fields.

Bifurcation Diagrams and Moduli Spaces of Planar Quadratic Vector Fields with Invariant Lines of Total Multiplicity Four and Having Exactly Three Real Singularities at Infinity

In this article we consider the class $${{\bf QSL}_{\bf4}^{\bf3s\boldsymbol\infty}}$$ of all real quadratic differential systems $${\frac{{\rm{d}}x}{{\rm{d}}t}=p(x,y),\frac{{\rm{d}}y}{{\rm{d}}t}=q(x,y)}$$ with gcd(p, q) = 1, having invariant lines of total multiplicity four and three real distinct infinite singularities. Firstly we construct compactified canonical forms for the class $${{\bf QSL}_{\bf4}^{\bf3s\boldsymbol\infty}}$$ so as to include limit points in the 12-dimensional parameter space of the set $${{\bf QSL}_{\bf4}^{\bf3s\boldsymbol\infty}}$$ . We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the moduli spaces under the action of the group of affine transformations and time homotheties and we place the phase portraits in these moduli spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under the group action.

On the number of limit cycles of planar quadratic vector fields with a perturbed center

On the number of limit cycles of planar quadratic vector fields with a perturbed center

Planar Quadratic Vector Fields with Two or Three Finite Singularities and a Finite Saddle Connection on a Straight Line

We classify the phase portraits of planar quadratic vector fields with a invariant straight line passing through two finite saddles, when the vector field has two or three finite singularities.

On the number of limit cycles of planar quadratic vector fields

On the number of limit cycles of planar quadratic vector fields

Planar Quadratic Vector Fields with Finite Saddle Connection on a Straight Line (Non-convex Case)

We classify the phase portraits of planar quadratic vector fields with a invariant straight line passing through two finite saddles, when the singularities of the field form a non-convex quadrilateral.

Planar quadratic differential systems with invariant straight lines of total multiplicity four

In this article we prove a classification theorem of real planar quadratic vector fields which possess four invariant straight lines, including the line at infinity and including their multiplicities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of algebraic invariants and comitants and also geometrically, using cycles on the complex projective plane and on the line at infinity. The algebraic invariants and comitants allow us to verify for any given real quadratic system whether or not it has invariant lines of total multiplicity four, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer. This classification is instrumental in studying the problem of Darboux integrability of such systems, work which is in progress.

Planar quadratic vector fields with finite saddle connection on a straight line (convex case)

Planar quadratic vector fields with finite saddle connection on a straight line (convex case)

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.

On the geometry in the neighborhood of infinity of quadratic differential systems with a weak focus

In this work we use the algebro-geometric concept of divisor on a projective curve in the study of planar quadratic vector fields. We introduce here specific divisors to encode global information about the geometry at infinity of quadratic systems with a weak focus which is not a center and we show that these concepts organise and unify in an intrinsic way this information. This geometric approach forms a link between chart-dependent classification studies of quadratic (or cubic) differential systems and affine-invariant results based on the algebraic theory of invariants of differential systems.

On the global analysis of the planar quadratic vector fields

On the global analysis of the planar quadratic vector fields

Quadratic systems of conservation laws with generic behavior at infinity

We identify quadratic systems of conservation laws with “generic” behavior at infinity, where the genericity conditions derive naturally when studying weak solutions of conservation laws. Namely, we identify quadratic models for which the vector field associated with the viscosity admissibility criterion has properties at infinity that are true for an open and dense subset of the set of all planar quadratic vector fields in the metric associated with the Euclidian space of coefficients. We determine the boundaries of the regions containing, “generic” models in the parameter space of coefficients of quadratic models. We show that when crossing the boundaries of nongeneric models transversally, the Poincare compactification of the corresponding vector field undergoes either a saddle node or a transcritical bifurcation at infinity. For quadratic models with a bounded elliptic region we calculate the loci of nongeneric models assuming the viscosity matrix to be the identity. We obtain a two-parameter normal form for such models and show that the boundaries that determine generic models in the two-dimensional parameter space correspond to the Schaeffer-Shearer classification of models with an isolated umbilic point. Since the loci of nongeneric models are invariant under the equivalence transformations that preserve weak solutions of conservation laws, understanding their behavior at infinity promises to provide an insight into a general classification of quadratic conservation laws.

A Class of Planar Quadratic Vector Fields with a Limit Cycle Surrounded by a Saddle Loop

A Class of Planar Quadratic Vector Fields with a Limit Cycle Surrounded by a Saddle Loop

A class of planar quadratic vector fields with a limit cycle surrounded by a saddle loop : Rodrigo Bamon, Instituto de Mathemȧtica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, CEP 22460 — Rio de Janeiro, RJ, BRASIL and Universidad de Chile, Santiago, CHILE.

A class of planar quadratic vector fields with a limit cycle surrounded by a saddle loop : Rodrigo Bamon, Instituto de Mathemȧtica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, CEP 22460 — Rio de Janeiro, RJ, BRASIL and Universidad de Chile, Santiago, CHILE.

A class of planar quadratic vector fields with a limit cycle surrounded by a saddle loop

Examples of planar quadratic vector fields with a limit cycle surrounded by a saddle loop are given by means of deformations of Hamiltonian vector fields.