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。     

一类五次微分系统的定性分析

对一类五次平面多项式微分系统进行了定性分析.给出原点的中心与等时中心条件及极限环的存在性.研究了此系统无穷远点的性态,该无穷远点是高次奇点,并运用把大角域分为若干小角域的方法对此高次奇点在不定号情形下轨线的分布情况进行讨论.     

一类2n+1次多项式微分系统的局部极限环分支

研究了一类2n 1次多项式微分系统在原点的局部极限环分支问题,通过计算与理论推导得出了该系统原点的奇点量表达式,确定了系统原点的中心条件以及最高阶细焦点的条件,并在此基础上构造出系统在原点分支出4个极限环的实例.     

一类七次多项式微分系统的中心条件与赤道极限环分支

研究了一类七次系统无穷远点的中心条件与赤道极限环分支问题.通过将实系统转化为复系统研究,给出了计算无穷远点奇点量的递推公式,并在计算机上用Mathematica推导出该系统无穷远点前14个奇点量,进一步导出了无穷远点成为中心的条件和14阶细焦点的条件,在此基础上得到了七次系统无穷远点分支出12个极限环的一个实例.     

一类五次多项式系统无穷远点等时中心条件与极限环分支

研究一类五次系统无穷远点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换将系统无穷远点转化成原点,然后求出该原点的前8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,在此基础上给出了五次多项式系统在无穷远点分支出8个极限环的实例.同时通过一种最新算法求出无穷远点为中心时的周期常数,得到了拟等时中心的必要条件,并利用一些有效途径一一证明了条件的充分性.     

一类高次奇点与无穷远点的中心焦点理论

对实平面微分自治系统论述了一类高次奇点与无穷远点的中心焦点判定、后继函数、形式级数、中心积分、积分因子、焦点量、奇点量以及极限环分支等问题.     

一类多项式向量场的中心条件

本文研究一类多项式系统的高次奇点和无穷远点的中心问题,对有限奇点（原点）和无穷远点（Poincare球面上的赤道）的中心问题进行统一处理,给出了系统原点和无穷远点为中心的一个充分条件。     

拟二次系统的广义焦点量与极限环分枝

本文给出了拟二次系统的前18个奇点量和可积性条件,由此统一解决了几类实平面微分自治系统的初奇点、高次奇点以及无穷远点的中心焦点判定与极限环分枝问题.     

一类具有13个参数的7次系统的极限环分支

研究了一类具有幂零奇点的7次多项式微分系统的极限环分支与中心问题.借助于数学软件MATHEMATICA,推导出系统在原点的前14个拟Lyapunov常数,从而得到了系统的原点为中心的充要条件,证明了系统在3阶幂零奇点处可以分支出14个极限环,给出了7次李雅谱诺夫系统在3阶幂零奇点处的环性数的下界.     

具有十个大振幅极限环的五次多项式系统

本文研究一类五次平面多项式系统赤道极限环分支问题.运用奇点量方法,首次证明了五次多项式系统可在赤道分支出十个极限环.     

CENTER CONDITIONS AND BIFURCATION OF LIMIT CYCLES FOR A CLASS OF FIFTH DEGREE SYSTEMS

The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated. Two recursive formulas to compute singular quantities at infinity and at the origin are given. The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles. Two fifth degree systems are constructed. One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity. The other perturbs six limit cycles at the origin.     

Center conditions and bifurcation of limit cycles at degenerate singular points in a quintic polynomial differential system

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.     

Bifurcations of limit cycles from infinity for a class of quintic polynomial system

In this work, we use an indirect method to investigate bifurcations of limit cycles at infinity for a class of quintic polynomial system, in which the problem for bifurcations of limit cycles from infinity be transferred into that from the origin. By the computation of singular point values, the conditions of the origin (correspondingly, infinity) to be the highest degree fine focus are derived. Consequently, we construct a quintic system with a small parameter and eight normal parameters, which can bifurcates 1 to 8 limit cycles from infinity respectively, when let normal parameters be suitable values. The positions of these limit cycles without constructing Poincaré cycle fields can be pointed out exactly.     

一类反应扩散方程的孤立周期波和局部临界周期分支

研究了一类含有五次非线性反应项和常数扩散项的反应扩散方程的小振幅孤立周期波解,以及它的行波方程局部临界周期分支问题.运用行波变换将反应扩散方程转换为对应的行波系统,应用奇点量方法和计算机代数软件MATHEMATICA计算出该系统的前8个奇点量,得到该系统奇点的两个中心条件,并证明行波系统原点处可分支出8个极限环,对应的非线性反应扩散方程存在8个小振幅孤立周期波解;通过周期常数的计算,得到了行波系统原点的细中心阶数,并证明该系统最多有3个局部临界周期分支,且能达到3个局部临界周期分支;通过分析行波系统的临界周期分支,得到该反应扩散方程有3个临界周期波长.     

The center conditions and bifurcation of limit cycles at the infinity for a cubic polynomial system

In this paper, the problem of center conditions and bifurcation of limit cycles at the infinity for a class of cubic systems are investigated. The method is based on a homeomorphic transformation of the infinity into the origin, the first 21 singular point quantities are obtained by computer algebra system Mathematica, the conditions of the origin to be a center and a 21st order fine focus are derived, respectively. Correspondingly, we construct a cubic system which can bifurcate seven limit cycles from the infinity by a small perturbation of parameters. At the end, we study the isochronous center conditions at the infinity for the cubic system.     

Bifurcation of limit cycles at the equator

This paper studies center conditions and bifurcation of limit cycles from the equator for a class of polynomial differential system of order seven. By converting real planar system into complex system, we established the relation of focal values of a real system with singular point quantities of its concomitant system, and the recursion formula for the computation of singular point quantities of a complex system at the infinity. Therefore, the first 14 singular point quantities of a complex system at the infinity are deduced by using computer algebra system Mathematica. What’s more, the conditions for the infinity of the real system to be a center or 14 degree fine focus are derived, respectively. A system of order seven that bifurcates 12 limit cycles from the infinity is constructed for the first time.     

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.     

Sixteen large-amplitude limit cycles in a septic system

In this paper, bifurcation of limit cycles from the infinity of a two-dimensional septic polynomial differential system is investigated. Sufficient and necessary conditions for the infinity to be a center are derived and the fact that there exist 16 large amplitude limit cycles bifurcated from the infinity is proved as well. The study relays on making use of a recursive formula for computing the singular point quantities of the infinity. As far as we know, this is the first example of a septic system with 16 limit cycles bifurcated from the infinity.