动力系统的可积、分支与嵌入流

项目名称： 动力系统的可积、分支与嵌入流

项目编号： No.11271252

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 张祥

作者单位： 上海交通大学

项目金额： 50万元

中文摘要： 本课题主要研究常微分方程的定性、分支和可积理论中几个相关的问题, 其中大多是没有解决的困难的公开问题. 这些课题是申请者研究工作的延续和深入. 具体研究内容如下：具有弱共振的无穷光滑和解析双曲微分同胚的无穷光滑和解析嵌入流的存在性, 以及unipotent微分同胚嵌入流的存在性. 解析可积微分系统在退化奇点邻域解析等价正规型的存在性, 以及广义解析可积微分系统的解析等价正规型的存在性; Darboux可积理论的进一步改进和推广. 平面多项式微分系统在dicritical情况下代数极限环的弱化Hilbert第16问题, 以及多项式Lienard微分系统代数极限环的存在性(这是Zoladek[Trans. Amer.Math.Soc.1998]没有很好解决的问题). 一维quaternion常微分方程(高阶Bernoulli方程和齐次方程)的整体动力学(不变环面和周期规的存在性及全局结构等).

中文关键词： 常微分方程；动力系统；可积理论；分支理论；嵌入流

英文摘要： This proposal mainly studies some related problems in the qualitative theory, bifurcation and integrability of ordinary differential equations, which mostly are the difficult open problems. These projects are the continuity of the applicant's past research. In details, the subjects to be studied in this project are the following. The existence of infinite smooth and analytic embedding flows of infinite smooth and analytic hyperbolic diffeomorphims with weakly resonances, and the existence of embedding flows of unipotent diffeomorphisms. The existence of analytic normalization of analytic integrable differential systems in a neighborhood of degenerate singularities, and of generalized analytic integrable differential systems; the further generalization and improvement of the Darboux theory of integrability. The weaken Hilbert 16th problem on algebraic limit cycles of planar polynomial differential systems in the dicritical case, and the existence of algebraic limit cycles of polynomial Lienard differential system(the problem that Zoladek did not solve in [Trans. Amer. Math. Soc. 1998]). The global dynamics(the existence of invariant tori and periodic orbits, and global structures) of one-dimensional quaternion ordinary differential equations(higher order Bernoulli equations and homogenous equations).

英文关键词： Ordinary differential equations；dynamical systems；integrability theory；bifurcation theory；embedding flow