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.

T.Ito 在以下两种情况下证明了 Poincare-Bendixson 型定理。情况 1：定义域为 D*^<2n>(1)={ZEC^n*SIGMA^^n__*Zi*^p*1}，其中p*3 为整数，全纯向量场的奇点为原点。情况 2：两个全纯向量场生成的完全可积系统，极簇的交集为起源。此外，他还构造了一些具有连通极簇的全纯向量场的例子。W.Matsumoto发现了具有系数的一阶系统的强双曲性条件。仅依赖于时间变量。S.Yotsutani发现了一种新的泊松方程数值计算方法。另一方面，他构造了一个新的变换，得到了 Sturm-Liouville 方程的范式。通过这种变换，他解决了一些开放性问题。H.Oka用康利指数法得到了一维常双曲慢流形情况下周期轨道或连接轨道存在的拓扑和代数条件。Y.Morita有稳定的充分条件圆盘中系数不均匀的 Ginzburg-Landau 方程的涡旋解。此外，他通过数值计算发现了扩散耦合可激励系统中的全纯分岔结构。

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・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の改良" 情報処理学会論文誌. 投稿中.

Morishita、Amano 和 Yotsuya：“使用代理电荷法改进泊松方程的数值计算”，日本信息处理学会会刊目前正在提交。

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

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

Tomas Gedeon, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka and James Reineck: "Conley index for fast-slow system I : One-dimensional slow variable" Journal of Dynamics and Differential Equations. (to appear).

Tomas Gedeon、Hiroshi Kokubu、Konstantin Mischaikow、Hiroe Oka 和 James Reineck：“快慢系统的康利指数 I：一维慢变量”动力学与微分方程杂志。