A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals

Christopher, Colin; Llibre, Jaume

doi:10.1090/s0002-9939-01-06253-0

Cited by 12 publications

(13 citation statements)

References 18 publications

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“…In that work all the invariant algebraic curves linear in the variable y, that is, defined by f (x, y) = p 1 (x)y + p 2 (x), where p 1 and p 2 are polynomials, are determined. The example appearing in [8] is a further study of an example appearing in [7] and the example given in [11] is also described in [7]. We show that all these quadratic systems, with an invariant algebraic curve of arbitrary degree can be constructed by the method explained in the previous section.…”

Section: Lemma 13mentioning

confidence: 85%

“…In [7] it is shown that, in case n ∈ N, an irreducible polynomial of degree n belonging to a family of orthogonal polynomials is a solution of equation (11). For instance, when Ω 1 (x) = 1, we get the Hermite polynomials, when Ω 1 (x) = x, we get the Generalized Laguerre polynomials and when Ω 1 (x) = 1 − x 2 , we get the Jacobi polynomials.…”

Section: Lemma 13mentioning

confidence: 99%

“…We have the linear differential equation (11) in the same notation as in Theorem 6 and we consider g(x, y) := L(x) − y A 2 (x) .…”

Section: Lemma 13mentioning

confidence: 99%

“…J. Moulin-Ollagnier [19] gives a family of quadratic Lotka-Volterra systems, each with an invariant algebraic curve of degree 2ℓ, where ℓ is the parameter of the family, without rational first integral. A simpler example is given by C. Christopher and J. Llibre in [11]. In [8] a family of quadratic systems with an invariant algebraic curve of arbitrarily high degree without a Darboux first integral nor a Darboux inverse integrating factor is given.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3 we consider all the families of quadratic systems with an algebraic curve of arbitrarily high degree known until the moment of composition of this paper and we show that they all belong to the construction explained in Section 2. The families of quadratic systems with an algebraic curve of arbitrarily high degree studied in this paper are the ones appearing in [7,8,11,19] and one example more first appearing in this work. This new example consists on a biparametrical family of quadratic systems, which we give an explicit expression of a first integral for, such that when one of the parameters is a natural number, say n, the system exhibits an irreducible invariant algebraic curve of degree n.…”

Section: Introductionmentioning

confidence: 99%

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Integrability of Planar Polynomial Differential Systems through Linear Differential Equations

Giacomini¹,

Giné²,

Grau³

2006

Rocky Mountain J. Math.

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In this work, we consider rational ordinary differential equations dy/dx = Q(x, y)/P (x, y), with Q(x, y) and P (x, y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function.We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families.We give an analogous method for constructing rational equations but by means of a linear differential equation of first order.2000 AMS Subject Classification: 34A05, 34C05, 34C20

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Section: Lemma 13mentioning

confidence: 85%

Section: Lemma 13mentioning

confidence: 99%

“…We have the linear differential equation (11) in the same notation as in Theorem 6 and we consider g(x, y) := L(x) − y A 2 (x) .…”

Section: Lemma 13mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integrability of Planar Polynomial Differential Systems through Linear Differential Equations

Giacomini¹,

Giné²,

Grau³

2006

Rocky Mountain J. Math.

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On the Darboux Integrability of Polynomial Differential Systems

Llibre

Zhang

2011

Qual. Theory Dyn. Syst.

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Abstract. This is a survey on recent results on the Darboux integrability of polynomial vector fields in R n or C n with n ≥ 2. We also provide an open question and some applications based on the existence of such first integrals. IntroductionIn many branches of applied mathematics, physics and, in general, in applied sciences appear nonlinear ordinary differential equations. If a differential equation or vector field defined on a real or complex manifold has a first integral, then its study can be reduced by one dimension. Therefore a natural question is: Given a vector field on a manifold, how to recognize if this vector field has a first integral defined on an open and dense subset of the manifold? In general this question has no a good answer up to now.In this survey we provide sufficient conditions for the existence of a first integral for polynomial vector fields in R n or C n with n ≥ 2. An open question which essentially goes back to Poincaré is presented. Finally some applications of the existence of this kind of first integrals to physical problems, centers, foci, limit cycles and invariant hyperplanes are mentioned. Darboux theory of integrabilitySince any polynomial differential system in R n can be thought as a polynomial differential system inside C n we shall work only in C n . If our initial differential system is in R n , once we get a complex first integral of this system inside C n the real and the imaginary parts of it are real first integrals. Moreover if that complex first integral is rational, the same occur for its real and imaginary parts. In short in the rest of the paper we shall work in C n .In this section we study the existence of first integrals for polynomial vector fields in C n through the Darboux theory of integrability. The algebraic theory of integrability is a classical one, which is related with the first part of the Hilbert's 16th problem [19]. This kind of integrability is usually called Darboux integrability, and it provides a link between the integrability of polynomial vector fields and the number of invariant algebraic hypersurfaces that they have (see Darboux [13] and Poincaré [36]). This theory shows the fascinating relationships between integrability (a topological phenomenon) and the existence of exact algebraic invariant hypersurfaces formed by solutions for the polynomial vector field. This theory is now known as the Darboux theory of integrability, see for more details the Chapter 8 of [16].2010 Mathematics Subject Classification. 34A34, 34C05, 34C14.

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Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

Gasull

Giacomini

2023

Qual. Theory Dyn. Syst.

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For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non-existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Liénard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.

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A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals

Abstract: Abstract. We give a class of quadratic systems without rational first integral which contains irreducible algebraic solutions of arbitrarily high degree. The construction gives a negative answer to a conjecture of Lins Neto and others.

Cited by 12 publications

References 18 publications

Integrability of Planar Polynomial Differential Systems through Linear Differential Equations

Integrability of Planar Polynomial Differential Systems through Linear Differential Equations

On the Darboux Integrability of Polynomial Differential Systems

Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

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