ON LIMIT CYCLES FOR A CLASS OF DIFFERENTIAL SYSTEMS WITH POSITIVE DEFINITE POLYNOMIAL

In this paper, the nonexistence, existence and the number of limit cycles for a class of differential systems with positive definite polynomial are considered, and the results obtained generalize and supplement those of [1].     

NEW FAMILIES OF CENTERS AND LIMIT CYCLES FOR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH HOMOGENEOUS NONLINEARITIES

We consider the class of polynomial differential equations x = -y+Pn(x,y), y = x + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. Inside this class we identify a new subclass of systems having a center at the origin. We show that this subclass contains at least two subfamilies of isochro-nous centers. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centers when we perturb them inside the class of all polynomial differential systems of the above form. In particular, we present a function whose simple zeros correspond to the limit cycles vvhich bifurcate from the periodic orbits of Hamiltonian systems.     

LIMIT CYCLES FOR A CLASS OF FIFTH-ORDER DIFFERENTIAL EQUATIONS

In this paper, we study a fifth-order differential equation. Using the averaging theory, we investigate the limit cycles of the equation.     

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.     

一类Z_2对称五次微分系统的中心条件和极限环分支

本文研究了一类Z2对称五次微分系统的中心条件和小振幅极限环分支.通过前6阶焦点量的计算,获得了原点为中心的充要条件,并证明系统从原点分支出的小振幅极限环的个数至多为6.最后通过构造后继函数,给出系统具有6个围绕原点的小振幅极限环的实例.     

一类平面微分系统极限环的存在唯一性

研究一类平面微分系统的极限环,利用Hopf分支理论得到了该系统极限环存在性与稳定性的若干充分条件,利用Л.A.Чepkac和Л.ИЖилевьыч的唯一性定理得到了极限环唯一性的若干充分条件.     

一个在无穷远点分支出八个极限环的多项式微分系统

本文研究一类高次系统无穷远点的中心条件与极限环分支问题．作者首先推出一个计算系统无穷远点奇点量的线性递推公式，并利用计算机代数系统计算出该系统在无穷远点处的前11个奇点量，从而导出无穷远点成为中心和最高阶细焦点的条件，在此基础上作者首次给出了多项式系统在无穷远点分支出8个极限环的实例。     

ON A CONJECTURE CONCERNING THE NON-EXISTENCE OF LIMIT CYCLES FOR QUADRATIC DIFFERENTIAL SYSTEMS

Aconjecturewasgivenin[l]asfollows:Ifforthesystemoftypeill(i.e.,6/0):wehavewhereW=(n b)(l n)'--a'(6 ZI n),thenthereisnolicestcycle(LC,forabbreviation)aroundO(0,0).Noticethatcondition(2)canbedividedintothefollowingfoursub-cases:i)m=0,a(b ZI)/0,n)a=0,m(l n)/0,iii)W=0,WI/0,tv)m--sa=2a2 n(l Zn)=0,a/0,WI/0.(3)Non-existenceofLCunderi)orn)hasbeenprovedin[2]515,Theorem15.1and15.2;andthatunderWI/0,6=0hasbeenprovedin[2]514.Ifiniii)m(l n)>0then[3]hasprovedthenon-existenceofLCaroundObytheDulacfun…     

ON THE NUMBER OF LIMIT CYCLES OF A POLYNOMIAL DIFFERENTIAL SYSTEM

In this paper, we employ qualitative analysis and methods of bifurcation theory to study the maximum number of limit cycles for a polynomial system with center in global bifurcation.     

极限环的表达式和计算

本文提出一种解析法和数值法相结合的方法，用来计算多项式微分系统的极限环，极限环表示为x=∑k≥0(ak cos kφ bk sin kφ),y=∑k≥0(ck cos kφ dk sin kφ)，先用解析法求出小参数时极限环的初始表达式，然后用增量法和迭代法求出任意参数时极限环满足给定精度的表达式，半稳定极限环和分叉值也可以计算。     

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.     

LIMIT CYCLES FOR A CLASS OF DIFFERENTIAL SYSTEM WITH POSITIVE DEFINITE POLYNOMIAL

In this paper we consider a class of differential systems with positive definite polynomial having exactly one and two limit cycles.Such a system is more extensive than paper[1,2].     

LIMIT CYCLES OF THIRD-ORDER DIFFERENTIAL EQUATION

In this paper, we investigate a third-order differential equation. Based on the averaging theory, we obtain sufficient conditions for the existence of periodic solutions to the equation.     

ON LIMIT CYCLES FOR A CLASS OF DIFFERENTIAL SYSTEMS WITH POSITIVE DEFINITE POLYNOMIAL

In this paper, the nonexistence, existence and the number of limit cycles for a class of differential systems with positive definite polynomial are considered, and the results obtained generalize and supplement those of [1].     

BOUNDEDNESS OF SOLUTIONS AND EXISTENCEOF LIMIT CYCLES FOR A NONLINEAR SYSTEMOF DIFFERENTIAL EQUATIONS

IntroductionTheautonomousplanarsystemoftheformwasfirststudiedbyLevinsonandSmithintheclassicalpaper[2]astheequivalelltsystellloftheequationofLienardtypeandlateronsomeauthorshavecontributedtotiletheoryofthissystemwithrespecttoqualitativebehaviorofsolutionsll--9].InthepresentpaperwestudyasystemOfaslightlymoregeneraltype,namely,wherethefllnctionsp(y),q(y),g(x)andh(x,y)arecontilluousforallvalueoftheirarguments,and3'resubjecttotheconditionswhichensurethattheexistenceofuniquesolutiontotheinitialvalu…     

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。     

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．     

FIVE LIMIT CYCLES TO A CLASS OF SYSTEMS IN R~3

In this paper, a bridge between near-homogeneous and homogeneous vector fields in R 3 is found. By the relationship between homogeneous vector fields and the induced tangent vector fields of two-dimensional manifold S 2 , we prove the existence of at least 5 isolated closed orbits for a class of n + 1 (n ≥ 2) systems in R 3 , which are located on the five invariant closed cones of the system.     

一类7次微分自治系统的极限环分支

研究一类平面7次微分系统,通过作两个适当的变换以及焦点量的仔细计算,得出了系统的无穷远点与2个初等焦点能够同时成为广义细焦点的条件,进一步得出在一定条件下该系统能够分支出15个极限环的结论,其中5个大振幅极限环来自无穷远点,10个小振幅极限环来自2个初等焦点.     

一类三次系统极限环的存在唯一性

本文得到三次系统x=-y(1-ax)(1-ax) δx-lx3,y=x(1-ax)(1-bx)极限环的存在性、唯一性及不存在性的完整结果．