Study of Perturbed Lotka–Volterra Systems Via Abelian Integrals

On the Number of zeros of the Abelian integrals for a Class of perturbed LiÉnard Systems

Modules of Abelian Integrals and Picard-fuchs Systems

We give a simple proof of an isomorphism between the two C[t]-modules: the module of relative cohomologies Λ/dH ∧ Λ and the module ofAbelian integrals corresponding to a Morse-plus polynomial H in two variables.Using this isomorphism, we prove existence and deduce some properties of thecorresponding Picard-Fuchs system. Department of Mathematics, Purdue University, West Lafayett...

Modules of the Abelian integrals and the Picard–Fuchs systems*

We give a simple proof of an isomorphism between two C(t)-modules corresponding to bivariate polynomial H with non-degenerate highest homogeneous part: the module of relative cohomologies 2/dH ∧ 1 and the module of Abelian integrals. Using this isomorphism, we prove the existence and deduce some properties of the corresponding Picard–Fuchs system. Mathematics Subject Classification: 14D05, 32S4...

LINEAR ESTIMATE OF THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR A KIND OF QUINTIC HAMILTONIANS

We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

Abelian integrals in holomorphic foliations

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singu...