Pseudo-abelian integrals on slow-fast Darboux systems

Abstract

We study pseudo-Abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.

Finite cyclicity of slow-fast Darboux systems with a two-saddle loop

“…Using the notations of [2], let P 0 = P 0 (x, y), P 1 = P 1 (x, y) be real bivariate polynomials and consider the differential system (2.1)…”

“…Clearly, these zeros are exactly the intersection points of the curves {Im D 1 = 0}, {Im D 2 = 0}, that is to say C 1 −π ∩ C 2 −π and C 1 +π ∩ C 2 +π . The intersection points have a transparent geometric meaning: they correspond to complex limit cycles intersecting the cross-section, or fixed points of the holonomy map Hol along the "figure height loop" which we recall now (see [6,5,2]). If the holonomy map Hol were analytic with respect to the parameters too, this would imply an uniform bound for the number of fixed points of Z(ε), when ε > 0 belongs to a sufficiently small neighborhood of the origin.…”

On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields

“…9 (i). The reader will recognize in the loop γ a key ingredient in the proof of the local boundedness of the number of zeros of pseudo-Abelian integrals in [3,4]. Although the result of Theorem 4 is existential, the proof we use leads to effective upper bounds on the number of the bifurcating limit cycles.…”

О Числе Предельных Циклов, Появляющихся При Возмущении 2-Седловых Циклов Векторных Полей На Плоскости

“…9(i). Читатель узнает в петле γ клю-чевую составляющую доказательства локальной ограниченности числа нулей псевдоабелевых интегралов в [3], [4].…”