Vector fields, invariant varieties and linear systems

The aim of this paper is to introduce a concrete notion of multiplicity for invariant algebraic curves in polynomial vector fields. In fact, we give several natural definitions and show that they are all equivalent to our main definition, under some "generic" assumptions.In particular, we show that there is a natural equivalence between the algebraic viewpoint (multiplicities defined by extactic curves or exponential factors) and the geometric viewpoint (multiplicities defined by the number of algebraic curves which can appear under bifurcation or by the holonomy group of the curve). Furthermore, via the extactic, we can give an effective method for calculating the multiplicity of a given curve.As applications of our results, we give a solution to the inverse problem of describing the module of vector fields with prescribed algebraic curves with their multiplicities; we also give a completed version of the Darboux theory of integration that takes the multiplicities of the curves into account.In this paper, we have concentrated mainly on the multiplicity of a single irreducible and reduced curve. We hope, however, that the range of equivalent definitions given here already demonstrates that this notion of multiplicity is both natural and useful for applications.

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“…The definitions and results of this section come from [Pereira 2001]. We state and prove simplified versions adapted to the complex plane.…”

Section: Algebraic Multiplicitymentioning

confidence: 99%

“…In [Pereira 2001] were introduced the extactic curves for polynomials vector fields. Essentially, these are curves of higher-order inflection points for a given vector field.…”

Section: Introductionmentioning

confidence: 99%

Multiplicity of invariant algebraic curves in polynomial vector fields

Christopher¹,

Llibre²,

Pereira³

2007

Pacific J. Math.

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181

221

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“…Statement (i) follows from the second part of Theorem 3 of Pereira [34] (see also Theorem 5.3 of [11] for dimension 2). Statement (ii) for dimension 2 (i.e n = 2) follows from [11] and for n > 2 it is proved in [28].…”

Section: On Darboux Integrability Of the Polynomial Differential Systmentioning

confidence: 99%

“…For a modern definition of the m-th extactic hypersurface and a clear geometric explanation of its meaning, the readers can consult Pereira [34]. Christopher et al [11] assuming the irreducibility of the invariant algebraic curves used the extactic curve to study the algebraic multiplicity of invariant algebraic curves of planar polynomial vector fields, and prove the equivalence of the algebraic multiplicity with other three ones: the infinitesimal multiplicity, the integrable multiplicity and the geometric multiplicity.…”

Section: Then It Has Dimensionmentioning

confidence: 99%

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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“…In general, the Darboux polynomial of thé system (1.1) can be found by solving the equation (1.2) for f and R f . Equation (1.2) is easy to solve if the degree of f is known in advance (for example, see [10,Proposition 1]). However, it is still an open problem, for a given system, to establish the upper bound for the degree of the invariant algebraic curve effectively.…”

Section: Introductionmentioning

confidence: 99%