非定常概周期系统局部结构稳定的充要条件

NECESSARY AND SUFFICIENT CONDITION FOR LOCAL STRUCTURAL STABILITY OF NONAUTONOMOUS ALMOST PERIODIC SYSTEMS

本文给出：如果一致概周期函数ｆ（ｔ，ｘ）满足：ｉ）ｆ（ｔ，０）＝０；ｉｉ）Ｄｘｆ（ｔ，ｘ）在Ｒ×Ｄ上有界（Ｄ是Ｒｎ中含原点的一个区域）；ｉｉｉ）对ｔ一致成立，那么系统关于Ｂ１（所有满足ｉ），ｉｉ）的一致概周期函数全体所成之集按范数构成的赋范线性空间）在原点邻域局部结构稳定的充要条件是它的零解变分系统在Ｒ上具有指数型二分性．进而得到系统关于Ｂ１在原点邻域局部结构稳定是可继承的．并推出这类结构稳定的系统在Ｂ１的某子空间中是处处稠密的．

In this paper,we establish the following results:For the uniformly almost periodic function t(t, x)if i) f(t, 0)=0;(II) Dxf(t,x) is bounded on R×D(D is a domain,0 D=Rn);iii) Dxf(t,0)= tim Dxf(t,x) uniformly forx→0t R,then,(1) the locally structural stability with respect to the space B1 of the differential system x'(t)= f(t, x) in a neighborhood of the orgin is equivalent that the variation system of zero solution has an exponential dichotomy on R;(2) in this case the structural stability has the inherited property; (3) the local structurally stable system in (1) is dense everywhere in a subspace of B1 whose member satisfies iii), where B1 is a normed linear space of uniformly almost periodic functions which satisfy i) and ii), with the norm B1=sup(t,x).