Una aplicación del teorema de liberalización de hartman

Abstract

In this note we consider the ordinary differential equation U' = AU +f(t, U), where A is a real, constant, n x n matrix with eigenvalues satisfying Reλ = 0. We assume that the solution U = 0 of the system U' =A U is stable. f is a C1 function, lim U→0 f(t, U) = 0 and lim U→0 Df(t, U) = 0 uniformly in t (Df is derivate of f with respect to U). We prove that for any μ and gt; 0, there exists δ (μ) and gt; 0 such that for solutions U1, U2 with initial conditions ξ1, ξ 2, ξ 1 "# ξ2, |ξ1| and lt; δ we havem limt→∞|U1 (t) – U2 (t)| eμt = ∞.