Аннотация:We obtain sufficient stability conditions for a system of ordinary differential equations of the form x ˙=f(t,x)+μR(t,x) with a small parameter μ under the assumption that the nonperturbed system has a critically stable zero solution. Stability ensures the sign definiteness of the integral of the product of the gradient of the Lyapunov function of the nonperturbed system and the vector of perturbations R(t,x) along some family of functions approximating in a certain sense the general solution of the nonperturbed system. We formulate an analogous statement in terms of vector Lyapunov functions.