Study on global property of holomorphic vector fields

利用孤立奇异点上的庞加莱-杜拉克多项式化定理，计算了与全纯向量场相关的2n-1维球面S^<2n-1>上的一维非奇异叶理的紧叶数。他证明了在系数仅依赖于时间变量的情况下，强双曲型系统可以在纤维空间的每一点上转化为具有对角主成分的Fuchsian。当x空间的维数为1时，这个条件是强双曲性的充分必要条件。另一方面，在高维情况下，他提出了变换矩阵对偶变量的均匀性的猜想，并证明了Petrovski的非强双曲系统是不稳定的，并且在一个小的扰动下很容易转变为强双曲系统。这是一个具有非线性边界条件的抛物型方程组。此外，他还研究了一类半线性椭圆型方程，得到了正径向解结构的一个强分类定理。该定理表明解的结构与由非线性项的系数和幂决定的函数的零点密切相关。h . oka证明了lorenz型混沌吸引子是由某个简并奇点的展开局部生成的。此外，她还研究了一些分段线性向量场，发现了包含无穷多个余维2同斜轨道的加倍的分岔，称为轨道翻转型。morita与h.n inoomiya和e.a yanagida一起得到了具有非线性边界条件的反应-扩散方程惯性流形存在性的定理。并给出了流形上的有限维简化方程及其在具体方程中的应用。这里的惯性流形是指一个有限维不变流形它吸引相空间中的每一个轨道。morita与K.Mischaikow共同研究了有界区间内金兹堡-朗道方程耗散动力系统的全局结构，并展示了动力学的摩尔斯结构。Y.Morita和S.Jimbo在环空区域构造稳定的非常解时，得到了任意凸域上Ginzburg-Landau方程非常平衡解的不稳定性定理。少

T.Ito counted the number of compact leaves of one dimensional non-singular foliation on the 2n-1 dimensional sphere S^<2n-1> associated with holomorphic vector fields, using Poincare-Dulac polynomialization Theorem at isolated singular point.W.Matsumoto has studied the characterization of the strongly hyperbolic systems of partial differential equations. He showed that in case where the coefficients depend only on the time variable, the strongly hyperbolic systems can be transformed to a Fuchsian with diagonal principal part at each point of the fiber space. This condition is the necessary and sufficient for the strong hyperbolicity when the dimension of x-space is one. On the other hand, in higher dimension case, he proposed a conjecture that the uniformity on the dual variables of transforming matrix might be necessary, and he showed that Petrovski's non-strongly hyperbolic system is unstable and easily change to a strongly hyperbolic system by a small perturbation.S.Yotsutani has st … More udied the asymptotic behavior and the rate of the convergence of solutions to a practical mathematical model of an chemical interfacial reaction, which is a parabolic system with nonlinear bondary conditions. Moreover, he has also studied some class of semilinear elliptic equations, and obtained a strong classification theorem of structure of positive radial solutions. The theorem claims that structure of solution is closely related with zeros of a function determined by coefficients and power of a nonlinear term.H.Oka showed that chaotic attractor of Lorenz-type is generated locally by an unfolding of some degenerate singularity. Also, she studied some piecewise linear vector fields and found a bifurcation including infinitely many doubling of codimension 2 homoclinic orbits which is called orbit-flip type.Y.Morita obtained with H.Ninomiya and E.Yanagida a theorem on the existence of an inertial manifold for a reaction-diffusion equation with nonlinear boundary condition. Moreover they gave the reduced finite-dimensional equation on the manifold together with applications to specific equations. Here the inertial manifold means a finite-dimensional invariant manifold which attracts every orbit in a phase space. Y.Morita studied in a joint work with K.Mischaikow a global structure of the dissipative dynamical system for the Ginzburg-Landau equation in a bounded interval and they showed a Morese structure of the dynamics. Y.Morita with S.Jimbo obtained an instability theorem of non-constant equilibrium solutions to the Ginzburg-Landau equation in any convex domain while they constructed stable non-constant solutions in an annulus domain. Less