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]。     

具有幂零奇点的七次Hamilton系统Abel积分的零点个数估计

本文研究了具有幂零奇点的七次Hamilton系统的Abel积分的零点个数问题.利用Picard-Fuchs方程法,得到了Abel积分I(h)=∮_(Γh)g(x,y)dx-f(x,y)dy在(0,1/4)上零点个数B(n≤3[(n-1)/4]),其中Γ_h是H(x,y)=x~4+y~4-x~8=h,h∈(0,1/4),所定义的卵形线f(x,y)=∑(1≤4i+4j+1≤n)aijx~(4i+1)y~4j)和g(x,y)=∑(1≤4i+4j+1≤n)bijx~4iy~(4j+1)是x和y的次数不超过n的多项式.     

一类二次可逆系统Abel积分的零点个数估计

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积分的零点个数估计,得出当n≥5时,上界为10[(4n+1)/3]+4[(4n)/3]+[(4n-1)/3]+13.     

十一次Hamilton系统的Abelian积分零点个数的线性估计

本文研究一类十一次Hamilton系统在n次多项式扰动下的Abelian积分的零点个数问题.利用Picard-Fuchs方程法,得到这类扰动Hamilton系统的Abelian积分的零点个数的上界(计重数).     

一类单中心二次可积系统的Abel积分零点个数

讨论了首次积分为H(x,y)=x~k(1/2y~2+Ax~2+Bx+C)的Abel积分的代数构造,并研究了k=2时具有一个中心的平面二次可积系统在n次扰动下的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.     

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．     

一类五次扰动Hamiltonian系统的Abel积分零点个数

研究一类五次扰动Hamiltonian系统的Abel积分零点个数上界.证明所研究的Abel积分的生成元构成精度为1的Chebeyshev系统,得到Abel积分零点个数上界是4(考虑零点重数).并指出前人文献中关于Abel积分零点个数上界的研究存在的错误,给出了最新结果.     

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.     

Perturbations from a kind of quartic Hamiltonians under general cubic polynomials

In this paper we investigate the perturbations from a kind of quartic Hamiltonians under general cubic polynomials. It is proved that the number of isolated zeros of the related abelian integrals around only one center is not more than 12 except the case of global center. It is also proved that there exists a cubic polynomial such that the disturbed vector field has at least 3 limit cycles while the corresponding vector field without perturbations belongs to the saddle loop case. This work was supported by National Natural Science Foundation of China (Grant No. 10671020)     

General Solution for Hamiltonians with Extended Cubic and Quartic Potentials

In terms of hyperelliptic functions, we integrate a two-particle Hamiltonian with quartic potential and additional linear and nonpolynomial terms in the Liouville integrable cases 1:6:1 and 1:6:8.     

Upper Bound of the Number of Zeros for Abelian Integrals in a Kind of Quadratic Reversible Centers of Genus One

By using the methods of Picard-Fuchs equation and Riccati equation, we study the upper bound of the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under polynomial perturbations of degree $n$. We obtain that the upper bound is $7[(n-3)/2]+5$ when $n\ge 5$, $8$ when $n=4$, $5$ when $n=3$, $4$ when $n=2$, and $0$ when $n=1$ or $n=0$, which linearly depends on $n$.     

Completeness of the Cubic and Quartic Henon-Heiles Hamiltonians

The quartic Henon-Heiles Hamiltonian passes the Painleve test for only four sets of values of the constants. Only one of these, identical to the traveling-wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the other three have not yet been integrated in the general case (α, β, γ) ≠ (0, 0, 0). We integrate them by building a birational transformation to two fourth-order first-degree equations in the Cosgrove classiffication of polynomial equations that have the Painleve property. This transformation involves the stationary reduction of various partial differential equations. The result is the same as for the three cubic Henon-Heiles Hamiltonians, namely, a general solution that is meromorphic and hyperelliptic with genus two in all four quartic cases. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painleve integrability (the completeness property).__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 14–25, July, 2005.