Poincare-Bendixson type theorem for holomorphic vector fields and its applications

全纯向量场的Poincare-Bendixson型定理及其应用

• 批准号: 09640137
• 负责人: ITO Toshikazu
• 金额: $ 1.6万
• 依托单位: Ryukoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640137/
• 关键词: holomorphic vector field; Poincare-Bendixson type theorem; Poisson equation; Sturm-Liouville equation; Ginzburg-Landau equation; homoclinic bifurcation; Conley index; slow-fast system; 非線型楕円型方程式; ポアソン方程式; ギンツブル・ランダウ方程式; 極集合; 偏微分方程式系; 南雲型Cauch

T.Ito proved Poincare-Bendixson type theorem in the following two cases.Case 1 : the domain is D*^<2n>(1)={ZEC^n*SIGMA^^n__*Zi*^p*1}, where p*3 is integer, and the singularity of holomorphic vector field is the origin.Case 2 : a completely integrable system generated by two holomorphic vector fields and the intersection set of polar varieties is the origin. Furthermore, he constructed some examples of holomorphic vector fields with connected polar varieties.W.Matsumoto discovered a condition of a strong hyperbolicity of first order systems with coeffictients. depending only on time variable.S.Yotsutani discovered a new method of numerical calculation of Poisson equation. On the other, he constructed a new transformation to get a normal form of Sturm-Liouville equation. By this transformation, he solved some open problems.H.Oka got topological and algebraic conditions of existence of periodic orbits or connecting orbits in the case of one dimensional normally hyperbolic slow manifold by Conley index method.Y.Morita had a sufficient condition of stable vortex solution of the Ginzburg-Landau equation with ununiform coefficient in a disc. Moreover, he discovered, by numerical calculation, a structure of holomorphic bifurcations in a diffusively coupled excitable system.

伊藤在以下两种情况下证明了庞加莱-本迪克森型定理。情形1：定义域为D*^<2n>(1)={ZEC^n*SIGMA^^n__*Zi*^p*1}，其中p*3为整数，全纯向量场的奇异点为原点。情形2：由两个全纯向量场生成的完全可积系统，其极变集为原点。在此基础上，他构造了具有连通极簇的全纯向量场的一些例子。松本发现了一阶带系数系统的强双曲性的一个条件。yotsutani发现了泊松方程数值计算的一种新方法。另一方面，他构造了一个新的变换，得到了Sturm-Liouville方程的标准形式。h . oka用Conley指数法得到了一维正双曲慢流形中周期轨道或连接轨道存在的拓扑和代数条件，y . morita给出了圆盘中具有非均匀系数的Ginzburg-Landau方程稳定涡解的充分条件。此外，他还通过数值计算发现了扩散耦合可激系统的全纯分岔结构。

项目成果

W.Matsumoto: "The strong hyperbolicity of first order system with coefficients depending only on time variable" International Conference on Differential Equations. to appear.

W.Matsumoto：“系数仅取决于时间变量的一阶系统的强双曲性”国际微分方程会议。

Y.Morita, M.Pernarowski and J.Dockery: "Homoclinic bifurcations in a diffusively coupled excitable system" Fields Institute Commucations. 21. 397-407 (1999)

Y.Morita、M.Pernarowski 和 J.Dockery：“扩散耦合可兴奋系统中的同宿分岔”菲尔兹研究所通讯。

森下・小林・高市・天野 四ツ谷: "代用電荷法によるポアソン方程式の数値計算" 情報処理学会論文誌. 38. 1483-1491 (1997)

Morishita、Kobayashi、Takaichi 和 Yotsuya Amano：“使用替代电荷法对泊松方程进行数值计算”，日本信息处理学会汇刊 38. 1483-1491 (1997)。

Waichiro Matsumoto: "The strong hyperbolicity of first order systems with coefficients depending only on time variable" International Conference on Differential Equations. (to appear).

Waichiro Matsumoto：“系数仅取决于时间变量的一阶系统的强双曲性”国际微分方程会议。

H.Kokubu, K.Mischaikow and H.Oka: "Directional transition matrix" Proceedings of the Conley Index Workshop. (to appear).

H.Kokubu、K.Mischaikow 和 H.Oka：“方向转换矩阵”康利指数研讨会论文集。