Trajectories of polynomial vector fields and ascending chains of polynomial ideals

Novikov, D.; Yakovenko, Sergeî

doi:10.5802/aif.1683

Cited by 40 publications

(57 citation statements)

References 23 publications

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“…Ce critère repose sur le concept de non oscillation qui suit. Signalons ici l'article [15] dédié à l'étude des solutions oscillantes de champs de vecteurs.…”

Section: Introductionunclassified

Solutions non oscillantes d’une équation différentielle et corps de Hardy

Blais¹,

Moussu²,

Sanz³

2007

Ann. inst. Fourier

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“…Ce critère repose sur le concept de non oscillation qui suit. Signalons ici l'article [15] dédié à l'étude des solutions oscillantes de champs de vecteurs.…”

Section: Introductionunclassified

Solutions non oscillantes d’une équation différentielle et corps de Hardy

Blais¹,

Moussu²,

Sanz³

2007

Ann. inst. Fourier

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“…The topological methods of continuation of the corresponding integral to the complex domain, which work so nicely in the Hamiltonian case, fail [12], thus the analytical continuation of the corresponding integral I(t) should be achieved by completely different methods. Besides, I(t) is not known to satisfy any reasonable linear or nonlinear differential equation of finite order, which makes impossible applications of the methods from [9,10]. For example, the local representation (1.5) already implies that I(t) is not a solution of a Fuchsian ordinary differential equation of finite order.…”

Section: Pseudoabelian Integralsmentioning

confidence: 99%

“…Other, completely different tools allow to place explicit upper bounds on the number of zeros of Abelian integrals which are at least δ-distant from the critical values of the Hamiltonian [9,10] (the bounds depend on δ > 0).…”

Section: 2mentioning

confidence: 99%

On Limit Cycles Appearing by Polynomial Perturbation of Darbouxian Integrable Systems

Novikov

2008

GAFA Geom. funct. anal.

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Abstract. We prove an existential finiteness result for integrals of rational 1-forms over the level curves of Darbouxian integrals.1. Limit cycles born by perturbation of integrable systems 1.1. Poincaré-Pontryagin integral. Limit cycles (isolated periodic trajectories) of polynomial planar vector fields can be produced by perturbing integrable systems which have nested continuous families of non-isolated periodic trajectories. The number and position of limit cycles born in such perturbations is determined by the number and position of isolated zeros of the Poincaré-Pontryagin integral of the dissipation form along the closed periodic orbits of the non-perturbed integrable vector field. The infinitesimal Hilbert problem is to place an upper bound for the number of isolated zeros of this integral. This problem was repeatedly formulated as a relaxed form of the Hilbert 16th problem by V. Arnold [1].Instead of polynomial vector fields, it is more convenient to deal with (singular) foliations of the real plane R 2 by solutions of rational Pfaffian equations θ = 0, where θ is a 1-form on R 2 with rational coefficients (since only the distribution of null spaces of the form θ makes geometric sense, one can always replace the rational form by a polynomial one). The foliation F is (Darbouxian) integrable, if the form is closed, dθ = 0. An integrable foliation always has a "multivalued" first integral of the form f (x, y) = exp r(x, y) · j p j (x, y) λ j , where r(x, y) is a rational function and p j polynomials in x, y, which are involved in the in general non-rational powers λ j . A particular case of the integrable foliations consists of Hamiltonian foliations defined by the exact polynomial 1-formIf L ⊂ R 2 is a compact smooth leaf (oval) of an integrable foliation θ = 0 belonging to the level curve {f = a} ⊂ R 2 , then all nearby leaves of this foliation are also closed and form a continuous family of ovals belonging to the level curves {f = t}, t ∈ (R 1 , a). The ovals from this family, denoted by L t , are uniquely parameterized by the values of the real variable t varying

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“…Besides, one can generalize the settings and consider arbitrary (not necessarily linear) foliations F defined by polynomial data and count isolated intersections of their leaves with affine subspaces of complimentary dimension. So far the problem is solved only for foliations of dimension 1 [NY99] and of codimension 1 [K91] or reducible to the latter case ("Pfaffian chains").…”

Section: Some Open Questionsmentioning

confidence: 99%