QUALITIVE THEORY OF THE QUADRATIC DIFFERENTIAL SYSTEMS (I)

Westudyinthispaperqualitativepropertiesofthequadraticdifferentialsystemsfundertheconditions:Inthissectionwewillshowthatunder(4),theanti-saddleFI(xl,yi)of(2)(withxl>0)lyingon1 ax--y=0canbeaweakfocusfortwotimesas6varIes.*ProjectsupportedbytheNationalNatural…     

ON TWO CONJECTURES OF THE QUADRATIC DIFFERENTIAL SYSTEMS

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A GENERAL PROPERTY OF THE QUADRATIC DIFFERENTIAL SYSTEMS

In this paper, it is proved that the quadratic differential systems with a weak saddle of order 2 or 3 have no closed or singular closed orbit. Then by the results of [3], it follows that the greatest order of the homoclinic loop bifurcation of a quadratic differential system is between 2 and 3 It means a homoclinic loop can be split into at most three limit cycles.     

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…     

ON THE NON-EXISTENCE OF LIMIT CYCLES OF CERTAIN QUADRATIC SYSTEMS

§１．Ｆｏｒｔｈｅｓｙｓｔｅｍｘ＝－ｙ＋δｘ＋ｌｘ２＋ｎｙ２＝Ｐ（ｘ，ｙ），ｙ＝ｘ（１＋ａｘ－ｙ）＝Ｑ（ｘ，ｙ），｛（１．１）ｗｅｃａｎｆｉｎｄｉｎ［１］ｔｈｅｆｏｌｏｗｉｎｇ：ＣｏｎｊｅｃｔｕｒｅＩ．Ａｓｓｕｍｅ１ａ＜０，ｎ＞１，ｎ＋ｌ＞０，ｎａ２...     

THE NUMBER OF LIMIT CYCLESOF THE PLANE AUTONOMOUS SYSTEMS

In this paper the author discusses the vector fields satisfying the conditions (C1) and (C2) and proves some theorems for the number of limit cycles of the plane autonomous systems. Some applications of the theorems and lemmas in §1 are given in §2.     

一类平面n+2次系统的定性分析

本文对一类具有实际意义的平面ｎ＋２次系统进行定性分析,得到了其极限环存在性、唯一性的较完整的大范围分析结果。     

A CLASS OF QUADRATIC HAMILTONIAN SYSTEMS UNDER QUADRATIC PERTURBATION

1 IntroductionConsider a Hamiltonian system with small perturbationwhere E is a small parameter. H(x, y), P(x, y) and Q(x, y) are all real polynimials of x and ywith degH = n 1, degP degQ 5 n. We suppose there is a fandly of ovals r(h) C {H(x, y) = h}for h E (ho, h1).The nunther of limit cycles of (1.1). l which tend to some r(h) as e -- 0, is closely relatedto the number of isolated zeroes of Abelian integraJThe next problem is to determille the lowest bound of the isolated zeroes of M…     

二次系统定性研究两种分类之间的关系及应用

本文建立二次微分系统实域定性研究Д．А．Черкɑс＾［１］的分类法与刘钧的分类法方程系数之间关系的公式。     

平面凸体的完全不等式组

对于平面凸体，分别用A、p、r、R、d表示它的面积、周长、内径、外径、直径．本文分别给出了其中的三元组合(A，r，R)，(p，r．R)，(p，r，d)，(A，r，d)所满足的不等式，并给出了相应的极值集．     

A FORMULA FOR INDEX OF A SINGULAR POINT OF POLYNOMIAL DIFFERENTIAL SYSTEMS IN THE PLANE

1IntroductionandConclllsionIntheplallalrPolynomialdifferentialsystem$~p.(x,y) of(x,y)=X(x,y),Z~q.(x,Y) '()(x,y)~I'(x,y)(l)wherep.(x,y),q.(x,y)arehomogenouspolynomialsinigyofthedegreen>1,(x,y)ER',withq:(x,7/) p:(x,y)/0,andof(x,y),,of(x,]j)arepolynomialinx,…     

BIFURCATION THEORY OF QUADRATIC DIFFERENTIAL SYSTEMS

This paper deals with the number of limit cycles and bifurcation problem of quadratic differential systems. Under conditions $a<0,b+2l>0,l+1<0$, the author draws three bifurcation diagrams of the system (1.18) below in the (\delta,m) plane, which show that the maximum number of limit cycles around a focus is two in this case.     

二次微分系统的反射函数及其周期解

本文给出了二镒多项式微分系统具有满足特定关系式的反射函数和存在周期解的充要条件，以及在此条件下反射函数的具体表达式及周期解的稳定性态。     

SOME DYNAMICAL PROPERTIESOF QUADRATIC RATIONAL MAPS

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二次滞后离散奇异系统的吸引域

本文利用Lyapunov方法对二次滞后离散奇异系统进行定性、定量分析并给出了该类系统的零解一致稳定吸引域和渐近稳定吸引域的定量估计.当初始扰动是在渐近稳定吸引域时,二次时滞离散广义系统的初始值问题的解趋于零.发现这些判据与滞后无关,容易测试,便于工程上的应用.通过示例描述了该方法的应用.     

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…     

GLOBAL EXISTENCE OF SOLUTIONS FOR QUADRATIC QUASI-LINEAR KLEIN-GORDON SYSTEMS IN ONE SPACE DIMENSION

Consider quadratic quasi-linear Klein-Gordon systems with eventually different masses for small, smooth, compactly supported Cauchy data in one space dimension.It is proved that the global existence holds when a convenient null condition is satisfied by nonlinearities.     

复杂系统中的自聚集,系统功能与正反馈

提出了复杂系统中自聚集的概念,这是广泛存在于复杂系统中的一种自组织的形式,然后考虑系统的功能,尝试对熟知的1+1＞2(总体大于部分之和),即关于系统功能的定性表达,寻求一种进一步定量定性的更一般的关系.给出了一个描述复杂系统的简单的网络模型,它有相当广泛的代表性,并用它分析系统的生长、演化及涌现过程以及在过程中功能的变化,得到一个定量定性的关系式f(n)=1/2n(n-1),表达功能变化规律.此式能揭示系统功能的一些性质并解释一些重要现象,诸如: 1+1＞2是此式当n=2时的特例,此式还给出具体的非线性;它还说明系统生长初期,组分的增减对系统的功能影响显著;当系统规模达到一定大时,将导致稳态涌现;以及系统功能从起始的脆弱性到稳态涌现的鲁棒性.f(n)还蕴涵着正反馈的激励放大机制,并给出一个表达式.据此说明此式可作为“报酬递增”的机制以及导致无标度网结构形成的“优先连接”的机制.最后是几句关于复杂与简单的话.