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

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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.