On Limit Cycles Appearing by Polynomial Perturbation of Darbouxian Integrable Systems

Abstract

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

Pseudo-abelian integrals on slow-fast Darboux systems

“…This paper is a part of the program of proving uniform finiteness of the number of zeros of pseudo Abelian integrals, see [7,2,3]. After studying the generic cases [7,2], nongeneric cases must be studied, too. Here we study zeros of pseudo-abelian integrals associated to deformations of slow-fast Darboux integrable systems.…”

“…To obtain an upper bound for increment of argument of I along the small arc a ε , we use an argument similar to those from [2,3,1,7]. One can easily prove that |I| C|t| M/ε in sectors.…”

Pseudo-abelian integrals: Unfolding generic exponential case

“…and η is a polynomial one-form of degree at most n. This integral appears as the linear term with respect to δ of the displacement function of a polynomial deformation (4) Mθ ε,α + δη = 0 of the Darboux integrable polynomial vector field with the first integral H ε,α , see ( [2] and [9]). Theorem 1.…”

“…The key of the proof [2,9] of the local boundedness of the number of zeros of a generic Darboux integrals on H = P a 1 1 · ... · P a k k P a k+1 k was a lemma stating that V ar a 1 ,··· ,a k ,a k+1 I(h) ≡ 0. The main result was then deduced from this by induction observing (via a generalization of Petrov's trick) that the operators V ar a reduce the number of isolated zeros of pseudo-abelian integrals by a constant locally bounded for any analytic family Θ Here Proposition 5 provides a suitable form of Petrov's trick.…”

Pseudo-Abelian integrals along Darboux cycles: A codimension one case

“…This is an analog of the Varchenko-Kchovanskii theorem for pseudoAbelian integrals. In the previous papers [1,7] the generic case was investigated. In this paper we consider a 1-parameter unfolding of the singular (non-generic) codimension 1 case.…”