Study on global property of holomorphic vector fields

利用孤立奇点上的Poincare-Dulac多项式定理，T.Ito计算了与全纯向量场相关的2n-1维球面S^&lt；2n-1&gt；上的一维非奇异叶面的紧叶个数。他指出，在系数只依赖于时间变量的情况下，强双曲系统可以变换为在纤维空间的每个点上具有对角主元的富奇线。当x-空间的维为1时，该条件是强双曲性的充要条件。另一方面，在高维情形下，他提出了一个猜想，即变换矩阵的对偶变量可能是一致的，并证明了彼得罗夫斯基的非强双曲系统是不稳定的，很容易通过小的扰动变成强双曲系统。S.Yotsutani有st…进一步研究了一类具有非线性边界条件的抛物型抛物型化学界面反应实用数学模型解的渐近性和收敛速度。此外，他还研究了一类半线性椭圆型方程，得到了正径向解的结构的一个强分类定理。该定理声称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些退化奇点的展开局部产生的。此外，她还研究了一些分段线性向量场，发现了一种包含无穷多个余维2同宿轨的倍增的分支，称为轨道翻转型。Y.Morita与H.Ninomiya和E.Yanagida得到了一个关于具有非线性边界条件的反应扩散方程的惯性流形的存在性定理。此外，他们还给出了流形上的约化有限维方程及其在具体方程中的应用。这里的惯性流形指的是吸引相空间中每个轨道的有限维不变流形。Y.Morita与K.Mischaikow在有界区间内研究了Ginzburg-Landau方程的耗散动力系统的全局结构，他们证明了动力学的Morese结构。Y.Morita和S.Jimbo得到了Ginzburg-Landau方程在任何凸域上的非常数平衡解的不稳定性定理，同时在环域上构造了稳定的非常数解。较少

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