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

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。

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.