Una aplicación del teorema de liberalización de Hartman

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 > 0, there exists δ (μ) > 0 such that for solutions U₁, U₂ with initial conditions ξ1, ξ 2,  ξ 1 "# ξ2, |ξ1| < δ we havem limt_→∞|U₁ (t) – U2 (t)| e^μ_t  = ∞.