Upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one

In this paper, by using the method of Picard-Fuchs equation and Riccati equation, we study the upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one under any polynomial perturbations of degree $n$, and obtain that their upper bounds are $3n-3$ ($n\geq 2$) and $18\left[\frac{n}{2}\right]+3\left[\frac{n-1}{2}\right]$ ($n\geq 4$) respectively, both of the two upper bounds linearly depend on $n$.     

A linear estimation to the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one

In this paper, using the method of Picard-Fuchs equation and Riccati equation, we consider the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under arbitrary polynomial perturbations of degree $n$, and obtain that the upper bound of the number is $2\left[{(n+1)}/{2}\right]+$ $\left[{n}/{2}\right]+2$ ($n\geq 1$), which linearly depends on $n$.     

一平面可积三次非Hamilton系统的Abel积分

本文讨论一平面可积三次非Hamilton系统在n次多项式扰动下Abel积分零点个数上确界，得到的结论是该Abel积分的零点个数的上确界为n。     

On the period function of reversible quadratic centers with their orbits inside quartics

This paper is concerned with the monotonicity of the period function for a class of reversible quadratic centers with their orbits inside quartics. It is proved that such a system has a period function with at most one critical point.     

一类双中心平面二次系统的Abel积分零点个数

利用Abel积分与第一、第二型完全椭圆积分,本文研究一类具有两个中心奇点的平面二次系统在n次小扰动下的Abel积分零点个数上界问题,得到了较小的上界估计．     

Bounding the number of zeros of certain Abelian integrals

In this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n Abelian integrals to have at most n+k−1 zeros counted with multiplicities. This condition involves the functions in the integrand of the Abelian integrals and it can be checked, in many cases, in a purely algebraic way.     

一类可积非Hamilton系统的Abel积分的零点估计

讨论了一类含参可积非Hamilton系统在一般二次多项式扰动下的Abel积分的零点,得出了不同参数范围下的Abel积分的零点数目的估计.     

一平面Hamilton系统的Abel积分的零点个数估计

本文讨论一平面Hamilton系统在一般n次多项式扰动下的系统的Abel积分的零点个数估计问题,得到的结论是:该系统的Abel积分的零点个数的上界为[(3n-1)/2]。     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

一类二次可逆系统的Abel积分

利用Picard-Fuchs方程法及Riccati方程法,研究了一类二次可逆系统在任意n次多项式扰动下Abel积分零点个数的上界问题,得到了当n≥4时,上界为10n+[n/2]-1.     

Monotonicity of the ratios of two Abelian integrals for Hamiltonian systems with parameters

We study the monotonicity of the ratios of two Abelian integrals $oint_{gamma_{i}(h)}ydx$ $backslash$ $oint_{gamma_{0i}(h)}xydx$ over three period annuli ${gamma_i(h)}$, for $i=1, 2, 3$, defined by a seventh-degree hyperelliptic Hamiltonian $H(x,y)=y^2+Psi(x)$ with a parameter. The parameter makes the problem more challenging to analyze. To over the difficulty, we apply some criterion with the help of transformations, tools in computer algebra such as boundary polynomial theory to determine the monotonicity of the ratios. Our results establish the existence and uniqueness of limit cycle bifurcated from each period annulus.     

一类可积非哈密顿系统的极限环个数的上界

In this paper, we consider the perturbations of two non-Hamiltonian integrable systems(1.3)μ, (4.1)μ. For the former,it is proved that the system under the polynomial perturbations hasat most f-n/2] limit cycles in the finite plane and the upper bound is sharp. The proof relies on acareful analysis of a related Abelian integral. For the latter, we obtain an estimate number of isolatedzeros of the corresponding Abelian integral.     

Monotonicity of the ratio of two Abelian integrals for a class of symmetric hyperelliptic Hamiltonian systems

In this paper we study the monotonicity of the ratio of two hyperelliptic Abelian integrals $I_0(h)=\oint_{\Gamma_h}ydx$ and $I_1(h)=\oint_{\Gamma_h}xydx$ for which $\Gamma_h$ is a continuous family of periodic orbits of a Newtonian system with Hamiltonian function of the form $H(x,y)=\frac{1}{2}{y^2}\pm \Psi(x)$, where $\Psi$ is an arbitrary even function of degree six.     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

Estimate of the number of zeros of Abelian integrals for a perturbation of hyperelliptic Hamiltonian system with nilpotent center

In this paper, we present a complete study of the zeros of Abelian integrals obtained by integrating the 1-form (α + βx + γ x²)ydx over the compact level curves of the hyperelliptic Hamiltonian of degree five $H(x\text{,}y)=\frac{y^2}{2}+\frac{1}{4}{x}^4-\frac{1}{5}{x}^5$. Such a family of compact level curves surround a nilpotent center. It is proved that the lowest upper bound of the number of the isolated zeros of Abelian integral is two in any compact period annulus, and there exists some α, β and γ such that system could appear at least two limit cycles bifurcating from the nilpotent center. The proof relies on the Chebyshev criterion for Abelian integrals (Grau et al, Trans Amer Math Soc 2011) and some techniques in polynomial algebra.     

Upper bounds for the number of zeroes for some Abelian integrals

Consider the vector field x^′=−yG(x,y),y^′=xG(x,y)$x^{\prime }=-yG(x,y),y^{\prime }=xG(x,y)$, where the set of critical points {G(x,y)=0}$\{G(x,y)=0\}$ is formed by K$K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n$n$ and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K$K$ and n$n$. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4$K\le 4$ we recover or improve some results obtained in several previous works.     

On the number of real quadratic fields with class number divisible by 3

We find a lower bound for the number of real quadratic fields whose class groups have an element of order . More precisely, we establish that the number of real quadratic fields whose absolute discriminant is and whose class group has an element of order is improving the existing best known bound of R. Murty.