On the Number of Zeros of Abelian Integrals for a Class of Quadratic Reversible Centers of Genus One

In this paper, using the method of Picard-Fuchs equation and Ric- cati equation, for a class of quadratic reversible centers of genus one, we re- search the upper bound of the number of zeros of Abelian integrals for the system (r10) under arbitrary polynomial perturbations of degree n. Our main result is that the upper bound is 21n - 24 (n ≥ 3), and the upper bound depends linearly on n.     

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.     

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 $n$.     

具有蝴蝶型相图的三次近Hamilton系统Abelian积分的零点个数

考虑了如下近Hamilton系统{x=2y(ax~2+2cy~2)+εf(x,y),y=2x(1-2bx~2)+εg(x,y),其中a0,c0,4bca~2,0|ε|■1,且f(x,y)和g(x,y)是关于x和y的3次多项式.得到了其相应Abelian积分孤立零点个数的上界.     

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 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.     

一类二次可逆系统Abel积分零点个数的上界

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积分零点个数的线性估计,得到了当n≥3时,上界为4[2n/3]+2[2n+1/3]+[2n+2/3]+16.     

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.     

亏格一双中心的二次可逆Lotka-Volterra系统的二次扰动

该文研究在二次扰动下,亏格一双中心的二次可逆Lotka-Volterra系统周期环域产生极限环的个数问题.证明在二次扰动下,二次可逆Lotka-Volterra系统(rlv5)的周期环域产生极限环的个数不超过3.     

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．     

一类Hamilton系统的Abel积分比值单调性的判别条件

用直接计算的方法对一类Hamilton系统的两个Abel积分比值的单调性进行讨论,指出该单词性条件可由两个判定函数直接确定．     

一类Hamiltion系统的Abel积分的零点个数估计

In this paper, we discuss the estimation of the number of zeros of the Abelian integral for the quadratic system which has a periodic region with a parabola and a straight line as its boundary when we perturb the system inside the class of all polynomial systems of degree n. The main result is that the upper bound for the number of zeros of the Abelian integral associated to this system is 3n-1.     

一类哈密顿系统的Abel积分之比单调性的判据

研究了一类哈密顿系统的两个Abel积分比值的单调性的条件,指出这个单调性条件可由文中给出的两个判定函数直接确定.     

一类二次可逆系统周期函数的单调性

利用Picard-Fuchs方程,研究了一类二次可逆系统周期函数的单调性问题,获得了在首次积分曲线是亏格1时的二次可逆系统周期函数单调的结论.     

一类四次Hamilton函数Abel积分零点个数的估计

证明了Abel积分I(h)=∮ΓhQ(x,y)dx-P(x,y)dy的零点个数的最小上界B(2n+2)=B(2n+1)≤3[n/2]+12[(n-1)/2]+4([p]表示P的整数部分),这里n是代数曲线H(x,Y)=x2士x4+Y4=h的连通闭分支,h∈E(Γh存在的最大开区间),P(x,y),Q(x,Y)是关于x,y 的次数不超过2n+2或2n+1的实多项式.     

Solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations at resonance

In this paper, we study the solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations with fractional derivative under resonant conditions. Firstly, the kernel function is presented through the Laplace transform and the properties of the kernel function are obtained. And then, some new results on the solvability for the boundary value problem are established by using Mawhin''s coincidence degree theory. Finally, two examples are presented to illustrate the applicability of our main results.