平面微分系统的中心问题与极限环分支

项目名称： 平面微分系统的中心问题与极限环分支

项目编号： No.10871206

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 轻工业、手工业

项目作者： 陈海波

作者单位： 中南大学

项目金额： 27万元

中文摘要： 本项目研究平面微分系统的中心焦点判定与极限环分支,在全高次的平面多项式微分系统有限远处奇点的中心焦点判定问题与弱分支函数、形式广泛的平面多项式微分系统的幂零奇点的奇点量计算的代数递推公式、三维非线性动力系统中心流形形式级数的新算法、奇点量与对应的焦点量的代数等价性、Z n - 等变向量场从细焦点分支出极限环、拟解析系统的中心问题与分支问题的研究，取得了较好的研究成果，同时，在微分方程周期解与同宿轨的研究上也获得了一些有意义的结果，已撰写发表论文多篇。平面微分系统的中心焦点判定与赤道极限环分支是微分方程定性理论及其应用研究的重要课题. 近年来，国内外学者围绕平面微分系统的定性研究做了大量工作，特别是美国、英国、加拿大及西班牙等国学者在中心条件与极限环的理论和应用研究方面很活跃。本项目的研究成果对于微分方程的理论创新及应用研究乃至相关学科的发展(如偏微分方程行波解、反应扩散方程的静态解分支等的研究)均具有极其重要的意义.

中文关键词： 平面微分系统；中心-焦点；极限环；分支

英文摘要： This project studies the determination of center-focus and bifurcation of limit cycles in planar differential systems.Achivements on determination of center-focus and weak bifurcation functions,algebric recursive formula of computation for the singular values of nilpotent singular points of high degree polynomial differential systems,a new algorithm of the formal series for the flow on center manifold of a class of three-dimensional nonlinear dynamic systems,equivalence of singular point values and focus values,limit cycles bifurcated from fine focus of Zq-equivariant vector fields,problem on center and bifurcation of quasi analytic systems are made.At the same time, some meaningful results on periodic solutions and homoclic orbits for differential equations are also obtained. Severial papers are published. The study on the determination of center-focus and bifurcation of limit cycles in planar differential systems with applications is an important theme in qualitative theory for differential equations. In recent years, scholars in the world have done much research on the qualitative theory of planar differential equations. In particular, scholars in United States, England, Canada and Spain are quite active on the study of center conditions and the theory of limit cycles with applications.Achivements on this project are of crucial importance for the theoretical innovation on differential equations with applications and the development of related subjects (such as the studies on travelling wave solutions of partial differential equations and the bifurcation of static solutions of reaction-diffusion equations).

英文关键词： Planar differential systems； Center-focus；Limit cycles； Bifurcation