Relationship between invariant sets of systems of differential equations and the corresponding difference equations

We study the relationship between invariant sets of systems of differential equations and the corresponding difference equations in terms of sign-constant Lyapunov functions. For systems of differential equations, we obtain a converse result concerning the existence of a positive-definite Lyapunov function whose zeros coincide with a given invariant manifold. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 280–285, April–June, 2006.