Quadratic Vector Fields Research Articles

PSI-SERIES OF QUADRATIC VECTOR FIELDS ON THE PLANE

Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields on the plane are considered. Its existence and convergence is studied, and an algorithm for the location of logarithmic singularities is developed. Moreover, the relationship between psi-series and non-integrability is stressed and in particular it is proved that quadratic systems with psi-series that are not Laurent series do not have an algebraic first integral. Besides, a criterion about non-existence of an analytic first integral is given.

Periodic orbits for a class of reversible quadratic vector field on [formula omitted

For a class of reversible quadratic vector fields on R 3 we study the periodic orbits that bifurcate from a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite part and another straight line at infinity in the local chart U 2 . More specifically, we prove that for all n ∈ N , there exists ε n > 0 such that the reversible quadratic polynomial differential system x ˙ = a 0 + a 1 y + a 3 y 2 + a 4 y 2 + ε ( a 2 x 2 + a 3 x z ) , y ˙ = b 1 z + b 3 y z + ε b 2 x y , z ˙ = c 1 y + c 4 z 2 + ε c 2 x z in R 3 , with a 0 < 0 , b 1 c 1 < 0 , a 2 < 0 , b 2 < a 2 , a 4 > 0 , c 2 < a 2 and b 3 ∉ { c 4 , 4 c 4 } , for ε ∈ ( 0 , ε n ) has at least n periodic orbits near the heteroclinic loop.

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.

Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

In this paper we consider vector fields in that are invariant under a suitable symmetry and that possess a 'generalized heteroclinic loop' formed by two singular points (e+ and e−) and their invariant manifolds: one of dimension 2 (a sphere minus the points e+ and e−) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e−). In particular, we analyse the dynamics of the vector field near the heteroclinic loop by means of a convenient Poincare map, and we prove the existence of infinitely many symmetric periodic orbits near . We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in , and the second one is the charged rhomboidal four-body problem.

Nonexistence of limit cycles for a class of structurally stable quadratic vector fields

The statistical analysis of the structurally stable quadraticvector fields made in [4] shows that the phase portrait 7.1(see Figure 1) appears without limit cycles, when the other threephase portraits in the same family with low probability sometimesappear with limit cycles. Here we prove that quadratic vectorfields having the phase portrait 7.1 have no limit cycles.

Phase portraits of the quadratic vector fields with a polynomial first integral

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral.

Polynomial first integrals of quadratic vector fields

We classify all quadratic polynomial differential systems having a polynomial first integral, and provide explicit normal forms for such systems and for their first integrals.

Quadratic vector fields in ℂP2 with solvable monodromy group at infinity

Quadratic vector fields for which the line at infinity is a phase curve with three different singular points are considered. It is assumed that the characteristic numbers of these singular points are not multiples of 1/4 or 1/6. It is shown that among the fields with fixed characteristic numbers satisfying this assumption, one can choose seven fields such that any other field with solvable noncommutative monodromy group at infinity is affine equivalent to one of the chosen fields. In addition, quadratic vector fields with commutative monodromy group at infinity are described.

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2

We classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 2. In other words we characterize all the global phase portraits of the quadratic polynomial vector fields having all their orbits contained in conics. For such a vector field there are exactly 25 different global phase portraits in the Poincaré disc, up to a reversal of sense.

Non-nested configuration of algebraic limit cycles in quadratic systems

This work deals with algebraic limit cycles of planar polynomial differential systems of degree two. More concretely, we show among other facts that a quadratic vector field cannot possess two non-nested algebraic limit cycles contained in different irreducible invariant algebraic curves.

Perturbations of non-Hamiltonian reversible quadratic systems with cubic orbits

This paper is concerned with degree n polynomial perturbations of a class of planar non-Hamiltonian reversible quadratic integrable system whose almost all orbits are cubics. We give an estimate of the number of limit cycles for such a system. If the first-order Melnikov function (Abelian integral) M 1 ( h ) is not identically zero, then the perturbed system has at most 5 for n = 3 and 3 n - 7 for n ⩾ 4 limit cycles on the finite plane. If M 1 ( h ) is identically zero but the second Melnikov function is not, then an upper bound for the number of limit cycles on the finite plane is 11 for n = 3 and 6 n - 13 for n ⩾ 4 , respectively. In the case when the perturbation is quadratic ( n = 2 ), there exists a neighborhood U of the initial non-Hamiltonian polynomial system in the space of all quadratic vector fields such that any system in U has at most two limit cycles on the finite plane. The bound for n = 2 is exact.

Semicompleteness of homogeneous quadratic vector fields

On étudie les champs de vecteurs holomorphes quadratiques et homogènes de C n qui sont semicomplets : ceux dont les solutions sont uniformes dans leurs domaines maximaux de définition. À un champ générique on associe de façon rationnelle quelques nombres complexes qui s’avèrent entiers dans le cas semicomplet. Ceci montre que, dans l’espace des classes d’équivalence linéaire de champs de vecteurs, les semicomplets sont contenus dans une sorte de réseau. On prouve que les feuilletages de CP n-1 induits par des champs quadratiques semicomplets sont linéarisables au voisinage de leurs points singuliers et on donne quelques familles nouvelles d’exemples dans C 3 . Finalement, on classifie les champs semicomplets de C 3 qui sont isochores et à singularité isolée.

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)

A note on the completeness of homogeneous quadratic vector fields on the plane

In this paper we give a full description of the complete homogeneous quadratic vector fields defined on the plane.

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.

Some criteria for the existence of limit cycles for quadratic vector fields

In this paper we consider real quadratic systems. We present new criteria for the existence and uniqueness of limit cycles for such systems by using Darbouxian particular solutions. Some results are based on the study of such systems in CP 2 . We also generalize the well-know result of Bautin on the nonexistence of limit cycles for quadratic Lotka–Volterra systems.

Periodic perturbations of quadratic planar polynomial vector fields

In this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. The global study envolving infinity is performed via the Poincaré compactification. The main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. It implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.

Finite cyclicity of elementary graphics surrounding a focus or center in quadratic systems

In this paper we prove that several elementary graphics surrounding a focus or center in quadratic systems have finite cyclicity. This paper represents an additional step in the large program to prove the existence of a uniform bound for the number of limit cycles of a quadratic vector field which we can call the finiteness part of Hilbert’s 16th problem for quadratic vector fields. It nearly finishes the part of the program concerned with elementary graphics. In [3] this problem was reduced to the proof that 121 graphics have finite cyclicity. The graphics considered here are the hemicycles (H 4 3 ), (H 5 3 ) and (H 6 3 ) together with (I 14a 2 ), (I 15b Emphasis>2 , (I 27 2 ) in the notation of [3] (Figure 1).

Sur les exemples de Lins Neto de feuilletages algébriques

We describe a family of quadratic homogeneous vector fields of C 3 having global meromorphic solutions that are not completely integrable. Under the standard projection, the orbits of these fields will give the degree two foliations of CP 2 recently constructed by Lins Neto in connection with the Poincaré problem. We will interpret these as quotients of linear flows on a product of elliptic curves. To cite this article: A. Guillot, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 747–750.

About a conjecture on quadratic vector fields

The Lotka–Volterra system of autonomous differential equations consists in three homogeneous polynomial equations of degree 2 in three variables. This system, or the corresponding vector field LV( A, B, C), depends on three non-zero (complex) parameters and may be written as LV( A, B, C)= V x ∂ x + V y ∂ y + V z ∂ z where V x=x(Cy+z), V y=y(Az+x), V z=z(Bx+y). As LV( A, B, C) is homogeneous, there is a foliation whose leaves are homogeneous surfaces in the three-dimensional space C 3 , or curves in the corresponding projective plane CP(2) , such that the trajectories of the vector field are completely contained in a leaf. An homogeneous first integral of degree 0 is then a non-constant function on the set of all leaves of the previous foliation. Trying to classify all values of the triple ( A, B, C) for which LV( A, B, C) has an homogeneous Liouvillian first integral of degree 0, we discovered the interesting family of Lotka–Volterra systems: SLV l=LV 2,B l=− 2l+1 2l−1 ,1/2 , l∈ N ★. This family provides a negative answer to the conjecture that there exists a uniform bound M 2 such that, if some homogeneous three-variable vector field of degree 2 has a particular solution (an irreducible Darboux polynomial in our words) of degree at least M 2, then the vector field is rationally integrable (has an homogeneous rational first integral of degree 0). Indeed: • SLV l has no homogeneous rational first integral of degree 0. • SLV l has an irreducible Darboux polynomial f l of degree m=2 l: (∗) x(y/2+z) ∂f l ∂ x +y(2z+x) ∂f l ∂ y +z(B lx+y) ∂f l ∂ z =((l−1)y+2lz)f l.