摘要
本文讨论n维空间上的向量场在其奇点附近的几何性质,主要结果如下列定理所述。定理考虑n维微分系统 x=X(x),x∈R^n (E) 设X∈C^1(R^n),X(0)=0,L=DX(0)。 (Ⅰ) 假定0为(E)的双曲奇点,则下列三点等价: (ⅰ) 矩阵L的所有特征根为实数; (ⅱ) 对(E)的任何解X(t)≠0,只要limx(t)=0(当t→+∞或-∞时),则极限limx(t)/||x(t)||存在; (ⅲ) 对x(E)的任何解x(t)≠0,只要limx(t)=0(当t→+∞或-∞时),则极限lim(x′(t)/||x′(t)||)存在。 (Ⅱ) 如果L的特征值均为实数且不为零,则有: 当limx(t)=0时,有lim(x(t)/||x(t)||=-lim(x′(t)/||x′(t)||); 当limx(t)=0时,有lim(x(t)/||x(t)||)=lim (x′(t)/||x′(t)||)
In this paper we prove a geometric property of orbits of ve ctor fields on R^n near hyperbolic singular points. Let f: R^n→R^n be an analytic function with f(O)=O, grad f=O at x=O. Then the grad f of f determines a vector field as follows dx_1/ dt=af/ax_1, dx_2/dt = Of/Ox_2, '',dx_n/dt =af/ax_n. suppose x(t)=(x_1(t)),…x_n(t)) is a solution of the above system with propcrty that limx(t)=O. The following problems were posed by profeSsor T-C.Kuo(University of Sydnoy) Probiem 1. Does x have a unique tangent at 0? That is, the li mit lim x(t)/x(t)=O exist? Problem 2. If the above limit exists, is it true that lim x′(t)/x′(t)=lim x(t)/x(t)?
