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 v...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)?展开更多
文摘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)?