THE PROBLEM OF LIMIT CYCLES FOR A LIENARD TYPE CUBIC SYSTEM

The purpose of this paper is to study a general Lienard type cubic system with one antisaddle and two saddles. We give some results of the existence and uniqueness of limit cycles as well as the evolution of limit cycles around the antisaddle for system (2) in the following when parameter a1 changes.     

具有三次曲线解的中心对称三次系统的极限环的存在性问题

本文证明了具有三次曲线解y＝αx3的中心对称三次系统可以存在极限环，从而纠正了文[1]认为具有三次曲线解的中心对称三次系统不可能存在极限环的错误结论     

一类三次系统极限环的个数与分布

本文研究一类三次系统的极限环,利用分支理论与定性分析技巧发现这类系统有四个极限环,并给出了他们的分布。     

GLOBAL BIFURCATIONS OF LIMIT CYCLES FOR A CUBIC SYSTEM

1MainResultsConsiderthecubicsystemwheree>Oissmall,parametersa=(a,,a,)EDCR2withDboundedandcl)cz)bl?bzareconstants.LetWesupposep'(y)2ofory/oorequivalentlyoscf53c2.(1.3)Assumethatg(x)hasonlysimplezeroswith6152/o.Withoutlossofgene-ralitywecansuppose610,5z<0(onecenterandtwosaddles);(3)5l-46z<0,bz>0(twocentersandonesaddle),Inthispaperweonlyconsiderthefirsttwocases.Obviously,(1'1).=ohasafamily0fperiodicorbitswhe…     

LIMIT CYCLES OF A QUADRATIC HAMILTONIAN SYSTEM BY CUBIC PERTURBATIONS

In this paper,we are concerned with a cubic near-Hamiltonian system,whose unperturbed system is quadratic and has a symmetric homoclinic loop.By using the method developed in [12],we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop.Further,we give a condition under which there exist 4 limit cycles.     

EXISTENCE OF LIMIT CYCLES FOR A CUBIC KOLMOGOROV SYSTEM WITH A PARABOLIC SOLUTION WHICH DOES NOT CONTACT AND INTERSECT THE COORDINATE AXES

This paper is concerned with a cubic Kolmogorov system with a parabolicsolution which does not contact and intersect the coordinates. The conclusionis that such a system may possess limit cycles.     

一类三次系统的极限环

本文讨论了一类三次系统 x=-y(1-αx~2)+δx-ιx~3,y=x(1-βx~2)和 x=-y(1-ax)(1-bx)+δx-ιx~3,y=x(1-cx)(1-bx)的极限环问题。     

一类三次系统的极限环个数与奇点分支

给出二次系统I的一类相伴系统在奇点O(0,0)的焦点量公式,证明了O至多为2阶细焦点,δlmn=0时系统在O外围至多有一个极限环,从而说明了系统在细焦点外围至多有一个极限环。最后给出了各个奇点的分支情况及几何特征。     

THE UNIQUENESS OF LIMIT CYCLE AND THE STRUCTURE OF CRITICAL POINT AT INFINITY FOR A CLASS OF CUBIC SYSTEM

A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.     

三次平面图中包含六点二边集的圈

得到了３－连通三次平面图具包含其给定六点二边集的圈的一个充分必要条件。并且列出一些悬而未决的研究问题。     

LIMIT CYCLES OF CUBIC VAN DER POL EQUATION WITH ONE FINITE CRITICALPOINT

ItiswellknownthatavarietyofsignificantpracticalproblemsleadtoVanderPciequation'Equation(l)canalsobewritteninitsequivalentformItisacubicdifferentialsystemiff(x,y)y h(x)isacubicpolynomial,correspondingly,equation(1)issaidtobecubicVanderPciequation.Inthispap…     

一类具有两个抛物线解的中心对称三次系统存在极限环的条件

本文给出了具有两个抛线解的中心对称三次系统存在在极限环的条件，它可能也是充要条件。     

LIMIT CYCLES OF QUARTIC AND QUINTIC POLYNOMIAL DIFFERENTIAL SYSTEMS VIA AVERAGING THEORY

We study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging theory. More precisely,we prove that the perturbations of the period annulus of the center located at the origin of a cubic polynomial differential system,by arbitrary quartic and quintic polynomial differential systems,there respectively exist at least 8 and 9 limit cycles bifurcating from the periodic orbits of the period annu...     

BIFURCATIONS OF A CUBIC SYSTEM PERTURBED BY DEGREE FIVE

In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving their two kinds of distributions (see Fig.5).     

奇点量公式的机器推导与一类三次系统的前10个鞍点量

本文给出了计算奇点量的两个递推公式，对一类三次系统用计算机推导出原点的前10个鞍点量，得到系统原点领域存在正则积分的充分必要条件。     

一类四次系统极限环的个数与分布

本文研究一类四次系统的极限环,通过计算四次系统鞍点分界线之间的有向距离,计算一阶焦点量 及二阶焦点量,判别同宿轨内外的稳定性,利用分支理论与定性分析技巧发现这类系统有六个极限环, 并给出了它们的分布．     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

THE E_3~1 TYPE OF CUBIC SYSTEMS WITH TWO INTEGRAL STRAIGHT LINES

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具三次曲线解的二次系统至多有一个极限环

本文研究具有三次曲线解x^3-x^2-y^2=0的二次系统,证明此类二次系统最多只有一个极限环,进而证明了具有三次的曲线解的二次系统至多有一个极限环。     

一类五次系统的赤道极限环问题

本文解决了一类五次系统赤道环的稳定性与极限环分枝问题,所得的结论与三次系统的若干结论形成有趣的对比.