微分系统奇点性质与分支的若干符号计算问题

The qualitative theory of differential equations is to meet the new development opportunity along with the computer algebra system's arising and improvement since the 1980's ..This project will focus on the following topics. We will study some symbolic calculation algorithm and reduction techniques related with polynomial operations such as Focus values, Lyapunov constants, Period constants etc. The center-focus determination and bifurcations of limit cycles for several classes of even degree polynomial differential systems will be discussed. Integrability and linearizability on p:-q resonant systems will be investigated. Center and isochronous center problems about quasi analytic systems and some computing problems on properties of nilpotent singular points will also be researched. The above problems are very significant in qualitative theory and bifurcation theory of differential equations..We will further research integrability for p:-q resonant systems of arbitrary degree, and investigate local bifurcations of critical periods on quasi-analytic systems. Many nonlinear wave equations can be turned into plane differential autonomic systems by some transformations. When there are bifurcations of limit cycles or local bifurcations of critical periods in a differential autonomous system under some perturbation, the dynamic behaviors of the corresponding nonlinear wave equation will also be considered. Aftermentioned problems are new and hardly seen in published literature..To study above problems will enrich the theory and the application results of ordinary differential equations, and promote the improvement of related subjects.

上世纪80年代以来,计算机代数系统的出现与推广给微分方程定性理论带来了新的发展契机。.本项目探讨焦点量、Lyapunov常数、周期常数等与多项式运算相关的若干符号计算算法和化简技巧， 利用符号计算研究几类偶数次多项式微分系统奇点的中心焦点判定和极限环分支问题，研究p:-q共振系统的可积性与可线性化问题，研究拟解析系统的中心、等时中心问题，研究与幂零奇点性质相关的几类计算问题。这些都是在微分方程定性理论和分支理论中有重要意义的问题。.进一步研究任意次p:-q共振系统的可积性问题；探索拟解析系统奇点的临界周期分支问题；许多非线波方程通过变换能化为平面微分自治系统，我们还将探索平面微分自治系统经扰动出现极限环或出现临界周期分支，其对应的非线性波方程动力学行为。后面几个问题在已有的文献中未被提出和研究过。.上述问题的研究将丰富常微分方程的理论和应用成果，促进相关学科的发展。

利用计算机符号计算为工具结合定性分析方法，研究平面微分系统的中心条件与极限环、p:-q 共振系统可积性和可线性化、拟解析系统幂零系统和Liénard系统的极限环与临界周期分支、高维系统的奇点量算法与中心流形上的极限环分支、非线性波动方程动力学性质等问题。获得了一系列的创新性成果，例如，四次系统原点和三次、四次Kolmogorov正平衡点的极限环数的最好结果，给出了一类(4,3)类型Liénard系统极限环的界，给出了研究一类任意次p:-q 共振系统可积性和可线性化的方法，得到了几类三维系统在中心流形上的极限环数目的最好结果，得到了非线性波方程与它的行波方程之间某些动力学性质的联系等等。. 对本项目的研究，我们共发表相关学术论文30篇，其中被SCI收录25篇。培养研究生9名，送培博士研究生2名。. 本项目所研究的问题是微分方程定性理论的重要课题，项目的研究成果对于微分方程的理论创新及应用具有重要的意义。