微分系统奇点性质与分支的若干符号计算问题

项目名称： 微分系统奇点性质与分支的若干符号计算问题

项目编号： No.11261013

项目类型： 地区科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 黄文韬

作者单位： 桂林电子科技大学

项目金额： 45万元

中文摘要： 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。

中文关键词： 符号计算；分支；p:-q 共振系统；拟解析系统；非线性波方程

英文摘要： 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,

英文关键词： symbolic computation；bifurcation；p:-q resonant system；quasi-analytic system；nonlinear wave equation