临界情形下的全局拓扑线性化

Global Topological Linearization in Critical Case

非线性系ｘ＇＝Ａｘ＝ｆ（ｘ）线性化的基本条件是Ａ的特征根实部异于零．经典线性化的结论只限于原点小邻域［１］．后改进为全局性的［２，３］，但要求ｆ（ｘ）有界．在文［４］中我们去掉了ｆ（ｘ）有界的限制．本文将进一步去掉Ａ的特征根实部异于零的限制，证明了，只要ｆ（ｘ）有适当结构，全局线性化仍是可能的．

The Basic condition to linearize a nonlinear system x' = Ax + f(x) is that none of the real parts of characteristic roots of A is zero. The classical conclusion of linearization is only limited to the small neighborhood of origion[1]. Later, in [2,3], it is generalized to global space, however, it required that f(x) must be bounded. In [4] we free the limitation. In this paper, we further free the limitation that none of the real parts of characteristic roots of A is zero and prove that the global linearization can be realized with a proper structure of f(x).