一类五次多项式系统的奇点量与极限环分支

该文研究一类五次多项式微分系统在高次奇点与无穷远点的极限环分支问题. 该系统的原点是高次奇点, 赤道环上没有实奇点. 首先推导出计算高次奇点与无穷远点奇点量的代数递推公式,并用之计算系统原点、无穷远点的奇点量，然后分别讨论了系统原点、无穷远点中心判据. 给出了多项式系统在高次奇点分支出5个极限环同时在无穷远点分支出2个极限环的实例. 这是首次在同步扰动的条件下讨论高次奇点与无穷远点分支出极限环的问题.     

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。     

一类多项式微分系统无穷远点的中心-焦点判定

本文研究了一类七次系统无穷远点的中心-焦点判定问题.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十二个奇点量,进一步导出了无穷远点成为中心的条件和分别成为七阶、九阶、十二阶细焦点的条件.     

论复自治微分系统的奇点量

本文研究复自治微分系统(E),得到如下的主要结果:(ⅰ)引入奇点量概念,实现了实系统中焦点量和鞍点量的统一;(ⅱ)定义Lie不变量概念,得到奇点量结构定理和广义对称原理;(ⅲ)计算了(E₃)的全部120个基本Lie不变量,并应用奇点量结构定理得到(E₂)和(E₃⁽³⁾)的奇点量公式及可积性条件。     

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.     

二次系统奇点的分支问题

本水文研究了(Ⅱ)和(Ⅲ)类的二次系统奇点的分支,并回答了[1]中提出的问题.     

一类五次系统的奇点量和可积性条件

本文采用代数运算方法研究了一类五次系统的原点奇点量和可积性条件，并给出了该系统的15个基本Lie-不变量。     

一类具有1:-4共振奇点的复三次Lotka-Volterra系统的可积性条件

对于一类具有1:-4共振奇点的复三次Lotka-Volterra系统,通过前12阶广义奇点量的计算,给出系统可积的充分条件.这些条件通过构造积分因子或形式积分得以证明.     

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.     

一类特殊五次系统的奇点量与可积性条件的研究

本文采用代数运算方法研究了一类五次系统的中心一焦点判定问题，给出了系统的13个基本如不变量，得到了直接用系统的系数表示的奇点量公式与可积性条件。     

Bifurcation of limit cycles at a nilpotent critical point in a septic Lyapunov system

In this paper, we characterize local behavior of an isolated nilpotent critical point for a class of septic polynomial differential systems, including center conditions and bifurcation of limit cycles. With the help of computer algebra system-MATHEMATICA 12.0, the first 15 quasi-Lyapunov constants are deduced. As a result, necessary and sufficient conditions of such system having a center are obtained. We prove that there exist 16 small amplitude limit cycles created from the third-order nilpotent critical point. And then we give a lower bound of cyclicity of third-order nilpotent critical point for septic Lyapunov systems.     

Center conditions and bifurcation of limit cycles at three-order nilpotent critical point in a cubic Lyapunov system

In the present paper, for the three-order nilpotent critical point of a cubic Lyapunov system, the center problem and bifurcation of limit cycles are investigated. With the help of computer algebra system-MATHEMATICA, the first 7 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact of there exist 7 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for cubic Lyapunov systems.     

一类具幂零奇点的五次系统中心条件及极限环分支

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of computer algebra system MATHEMATICA, the first 8 quasi Lyapunov constants are deduced. As a result, the necessary and sufficient conditions to have a center are obtained. The fact that there exist 8 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems.     

p-q-Laplace系统周期解的存在性

利用临界点理论研究了p-q-Laplace系统的周期边值问题.首先定义p-q-Laplace系统的弱周期解;其次给出一些引理;然后用临界点理论中的极小极大方法得到关于p-q-Laplace系统弱周期解的一个存在性定理;最后讨论了p-q-Laplace系统的相关问题.本文使用的主要方法是临界点理论中的环绕定理.     

On the equivalence of singular point quantities and the integrability of a fine critical singular point

In this paper, the equivalence of singular point quantities and the integrability of a fine critical singular point are discussed. For an extensive class of complex autonomous differential systems, the theoretical basis for the methods of using formal series and integrating factors to calculate the singular point quantities for the origin (a fine critical singular point) and judge its integrability is obtained. Two recursion formulas, which are much simpler than focal values for computation because the operation is rational, for computing singular point quantities are introduced.     

Multiplicity results for some elliptic systems near resonance with a nonprincipal eigenvalue

Some multiplicity results of solutions are obtained for a class of elliptic systems which are near resonance with a nonprincipal eigenvalue by the classical saddle point theorem and a local saddle point theorem in critical point theory.     

Center conditions and bifurcation of limit cycles at three-order nilpotent critical point in a seventh degree Lyapunov system

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of seventh degree system are investigated. With the help of computer algebra system MATHEMATICA, the first 12 quasi Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 12 small amplitude limit cycles created from the three order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for seventh degree Lyapunov systems.     

Limit cycles in a class of switching system with a degenerate singular point

Although switching systems have been investigated intensively, there are few results about limit cycles bifurcated from switching systems with degenerate singular point. In this paper, a method to compute focal values for degenerate critical point of switching systems was proposed. Furthermore, we studied a quartic system in order to illustrate the efficiency of our method.     

Bifurcations of limit cycles in a quintic Lyapunov system with eleven parameters

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of computer algebra system MATHEMATICA, the first 12 quasi Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 12 small amplitude limit cycles created from the three order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems, the result of Jiang et al. (2009) [18] was improved.     

Some existence results on periodic solutions of ordinary p-Laplacian systems

Some existence theorems are obtained for periodic solutions of ordinary p-Laplacian systems by the minimax methods in critical point theory.