The Algebraic Critical Cycle and Bifurcation Function of Limit Cycle for the System =xy, =sum from 0≤j+i≤2 (a_(ij)x~iy~j)

In the quadratic system possessing one integral straight line (?)=xy,(?)=sum from o≤i+j≤2 (a_(ij)x~iy~j) (1) there may exist a triangle critical cycle which is formed of two rays and a section of arc of equator on Poincar sphere, if(1) possesses two integral straight lines. Besides, in that paper the bifurcation function of limit cycle of(1) is also found. In this article we prove that there may exist a lune of critical cycle which is formed of an algebraic curve and a section of arc of the equator on the Poincar sphere; or is formed of an algebraic curve and a straight line. Moreover, we will also find out bifurcation function of limit cycle.     

ON EQUILIBRIA AND PERIODIC ORBITS OF COOPERATIVE SYSTEMS IN R~3

In the paper we present new criteria for existence of equilibria and periodic orbits for 3-dimensional systems which are cooperative. Particular attention is given to the relations of equilibria, cycle K and stable manifold of K. Meanwhile, the results of Hirsch [1, Theorem 9] and Jiang [2] are improved.     

POINTWISE ESTIMATE FOR ALGEBRAIC POLYNOMIAL APPROXIMATION

Let f(x)∈C[-1, 1], and ∏_n be the class of algebraic poly-nomials of degree n. It is well known that algebraic polynomialapproximation can be improved near the end points of the interval[-1, 1]. This fact was first observed by A. F. Timan, and laterJ. Brudnyi proved that for any positive integer k, there is aρ_n(x)∈∏_n such that     

Some extremal properties of the integral of legendre polynomials

1. Introduction and Main ResultsDenote by Pn the set of algebraic polynomials of degree not exceeding n. LetX = {X = (xl,xz,...,x.). 1 = xl > xz >' > xtL--l > xtL = --l}, n 2 2and let for X E X.Erd6s in [21 raised the question of determining Y E X such thatReceived July 21, 1999.also conjectured that Y = Z satisfyingw,,(Z, x) = c1' l" Pt.--1 (x) dx,j-- 1 Pt.-- 1 (x) dx,1 .2>where P,--1 stands for the (n -- 1)th Legendre polynomial normalized by Pn--1 (1) = l. Thiscolljecture was di…     

QUADRATIC SYSTEMS WITH A 3RD-ORDER (OR 2ND-ORDER) WEAK FOCUS

1 IntroductionSince a quadratic system has no limit cycle around a 3rd-order weak focu,[1]and has at most one limit cycle surrounding a 2nd-order weak fOcus['], study-ing the number of limit cycles of a p1anar quadratic system with a 3rd-order(or 2nd-order) weak focus we only need to study the number of limit cyclessurrounding the strong focus for the system. Without loss of generality thequadratic system with a 3rd--order (or 2nd--order) weak foclls and a strong focuscan be written in the fo.…     

An Extension of the Turan Inequality in L_p-space for 0

周颂平 《数学研究与评论》1986,(2)

HOMOCLINIC CYCLES OF A QUADRATIC SYSTEM DESCRIBED BY QUARTIC CURVES

This paper is devoted to discussing the topological classification of the quartic invariant algebraic curves for a quadratic system. We obtain sufficient and necessary conditions which ensure that the homoclinic cycle of the system is defined by the quartic invariant algebraic curve. Finally, the corresponding global phase diagrams are drawn.     

A SUFFICIENT AND NECESSARY CONDITION OF THE EXISTENCE TO A DOUBLE BUN TYPE SEPARATRIX CYCLE FOR AXIAL SYMMETRY CUBIC SYSTEM

1 IntroductionThere is almost no other type of algebraic curve which may be able tofOrm a separatrix cyc1e containing a fOcus fOr quadratic systern except para-bola, hyperbo1a, ellipse, parabola and straight line, two half lines and relativeequatorial arc, a class of cubic curve and two classes of quartic curve (see [1]for details). However, besides of these, there are many other types of algebraiccurves which may be able to fOrm a separatrix cycle fOr cubic system, fOr ex-ample, double foli…     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.     

三次系统的一类三曲线分界线环

本文证明了三次系统不可能同时存在三个三曲线x^3-3xy^2-1=0分界线环,但是可以同时存在两个三曲线分界线环,给出了同时存在两个三曲线分界线环的充要条件。     

A NECESSARY AND SUFFICIENT CONDITION FOR THE COEXISTENCE OF A CLASS OF CUBIC CURVE SEPARATRIX CYCLES AND LIMIT CYCLES TO THE CUBIC SYSTEM

In this paper, we give the necessary and sufficient condition for the coexistence of a class of cubic curve separatrix cycles and limit cycles to the cubic system, and study their topological structures.     

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。     

中心对称三次系统存在双曲线分界线环的充要条件

本文给出了中心对称三次系统存在双曲线分界线环的充要条件，并证明了此系统还可以至少存在五个极限环。     

三次系统E_3~1存在四角形双曲线分界线环的充要条件

得到了三次系统Ｅ１３存在四角形双曲线分界线环的充要条件，并给出了它们的拓扑分类和各种拓扑结构的参数条件．     

三次系统E_3~1存在二角形双曲线分界线环的充要条件

得到了三次系统 E1 3存在二角形双曲线分界线环的充要条件 .并给出了它们的拓扑分类和各种拓扑结构的参数条件 .     

Cycles with Prescribed and Forbidden Sets of Elements in Cubic Graphs

 A necessary and sufficient condition for the existence of a cycle containing a set of three vertices and an edge excluding another edge is obtained for 3-connected cubic graphs by means of canonical contractions. A necessary and sufficient condition is also obtained for a cyclically 4-connected cubic graph to have a cycle that contains a set of four vertices and that avoids another vertex. Received: October 4, 1999 Final version received: June 14, 2000     

由控制多边形内部两点确定的三次有理Bézier曲线

１．引言主要应用于自由曲线设计的有理Ｂｅｚｉｅｒ曲线在ＣＡＧＤ中起了重要作用．有理Ｂｅｚｉｅｒ曲线的几何形状不仅受其控制多边形而且受其权因子的控制,有关这方面的研究正受到越来越多的关注,例如［１－７］．当控制多边形给定时,权因子为有理Ｂｅｚｉｅｒ曲线的形状控制提供了自由度．权因子的性质及其与有理Ｂｅｚｉｅｒ曲线形状的关系较为复杂,目前尚未得到全面研究·文［４,５］给出了当修改有理Ｂｅｚｉｅｒ曲线上的一点时,权因子的计算公式,但该公式不能用于同时修改曲线上两点的情况,从而限制了修改曲线的灵活性．文…