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.     

A SPECIAL COMPLETE HYPER-ELLIPTIC INTEGRAL OF THE FIRST KIND

In this paper, we study a class of complete Abelian integral. We give an exact number and the upper bound of the number of zero points of the integral under some conditions.     

LIMIT CYCLE BIFURCATIONS OF NON-SMOOTH NEAR-HAMILTONIAN SYSTEM

In this paper,using the Abelian integral,we investigate the limit cycle bifurcation of two classes of non-smooth near-Hamiltonian systems,and obtain the maximum number of limit cycles of the system.     

A spectral interpretation for the explicit formula in the theory of prime numbers

A spectral interpretation for the poles and zeros of the L-function of algebraic number fields is given by Meyer. As Meyer works with Schwartz spaces which are not Hilbert spaces, the information on the location of zeros of the L-function is lost. In 1999, A. Connes gave a spectral interpretation for the critical zeros the Riemann zeta function. He works with Hilbert spaces. In this paper, we show that a variant of Connes’ trace formula is essentially equal to the explicit formula of A. Weil.     

Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

In this paper,we consider Li′enard systems of the form dx/dt=y,dy/dt=x+bx3-x5+ε(α+βx2+γx4)y,where b∈R,0|ε|1,(α,β,γ)∈D∈R3 and D is bounded.We prove that for |b|1(b0) the least upper bound of the number of isolated zeros of the related Abelian integrals I(h)=∮Γh(α+βx2+γx4)ydx is 2(counting the multiplicity) and this upper bound is a sharp one.     

一类通有二次哈密顿系统的环性(英文)

1 IntroductionConsider the fOllowing system:t'here E is a small parameter, H(I, y) is a polynomial of degree (n + 1 ) ? f(I, y)and g(x, y) are polynomials of degree S n. The corresponding Abelian integrallswhere b(h) is a compact leve1 curve H--'(h) of (1.l) when E = 0, for h lyingbetween criticaI values of H. The Hilbert-Arnold prob1em (weakend Hilbert16th problern) is to find an upper boulld of the nurnber of zeros of (l.2) forfixed n 2 2 and for al1 H, f and g. In this paper, we tvi11…     

高阶抛物不等式组的Liouville型定理及其应用

In this paper, we establish a Liouville-type theorem for a system of higher-order parabolic inequalities by using the method of test functions and an integral estimate. As an application, we observe the Fujita blow-up phenomena for the corresponding parabolic system, which in particular fills up the gap in the recent result of Pang et. al. （Existence and nonexistence of global solutions for a higher-order semilinear parabolic system, Indiana Univ. Math. J., 55（2006）, 1113-1134）. Moreover, the importance of this observation is that we do not impose any regularity assumption on the initial data.     

NEW OSCILLATION CRITERIA RELATED TO EULER S INTEGRAL FOR CERTAIN NONLINEAR DIFFERENTIAL EQUATION

Using the integral average technique and a new function,some new oscillation criteria related to Euler integral are obtained for second order nonlinear differential equations with damping and forcing. Our results are of a higher degree of generality than some previous results. Information about the distribution of the zeros of solutions to the system is also obtained.     

剩余为几乎2-正则的6-圈系

Let Kn be a complete graph on n vertices. In this paper, we find the necessary conditions for the existence of a 6-cycle system of Kn - L for every nearly 2-regular leave L of Kn. This condition is also sufficient when the number of vertices of L is n - 4.     

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。     

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.     

Abelian integrals and limit cycles

The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.     

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,     

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.     

On the number of limit cycles bifurcating from a non-global degenerated center

We give an upper bound for the number of zeros of an Abelian integral. This integral controls the number of limit cycles that bifurcate, by a polynomial perturbation of arbitrary degree n, from the periodic orbits of the integrable system , where H is the quasi-homogeneous Hamiltonian H(x,y)=x2k/(2k)+y²/2. The tools used in our proofs are the Argument Principle applied to a suitable complex extension of the Abelian integral and some techniques in real analysis.