On the number of zeros of Abelian integrals

Abstract

Abstract. We prove that the number of limit cycles generated from nonsingular energy level ovals (periodic trajectories) in a small nonconservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing infinitesimal Hilbert 16th problem.The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection defined over Q (the Gauss-Manin connection) with a quasiunipotent monodromy group.

On the number of limit cycles near a homoclinic loop with a nilpotent singular point

“…However, except for a very limited number of cases, the precise value of Z(m, n) is not known. More detailed information about the weakened Hilbert 16th problem can be found in [4,13,16,33,37,45].…”

“…In other words, this aspect mainly concentrates on finding as many as possible limit cycles by choosing suitable parameters, see for example [1,8,9,25,26,39,41]. The other is to study the maximal number of limit cycles and give an upper bound of it, see [4,19,32,49,52,54,55] and the references therein. The problem we will study in this paper belongs to the first case.…”

Quasialgebraic functions

“…Theorem 2.13 ( [BNY10]). If a quasiunipotent flat regular Pfaffian system (6) is defined over Q and has complexity at most s ∈ N, s 2, then…”

“…This construction was produced in [BNY10] for the particular case where the polar locus {Z = 0} of the forms is a fixed line in CP 2 so that the space of all forms is a linear space of polynomial 1-forms in a fixed affine chart. The general case will be treated elsewhere.…”

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

“…6]. Этот метод был использован в более общем контексте и в других работах, например, в [3], [2], а в [10] он исполь-зовался для того, чтобы заменить абелевы интегралы отображением первого возвращения Пуанкаре.…”