As is well known [1-4], the aim of the reduction principle is to present conditions enabling one to replace the stability problem concerning the entire space by a stability problem with respect to a closed subspace. At first glance, it seems that the reduction principle is trivial for the instability property. However, here we can single out a class of systems for which the problem of reducing the property of instability to its consideration on a subspace has a nontrivial meaning. Let a system of differential equationsformulabe given, where G is an open neighborhood of the origin and f : G £ R+ ! Rn is a continuous function. Suppose that f(x; t) satisfies the Lipschitz condition with respect to x and for any pair (x0; t0) 2 G£R+; x(x0; t0; t) is a solution of (1) such that x(x0; t0; t0) = x0. Let L+(x0; t0) is positive limit set of x(x0; t0; t); B" = fx 2 Rn : jjxjj < "g; " > 0; and d(x; y) = jjx ¡ yjj 8x; y 2 Rn. We formulate now the reduction principle for equilibrium instability. Theorem. Let Y be a closed subset of G and U be a neighborhood of Y . Suppose that, for system (1), there exists a function V 2 C1(U £R+;R+), such that for all (x; t) 2 U £ R+ the following conditions hold:formulaThen the zero solution of the system (1) is unstable. Here _V (x; t) is the derivative of V (x; t) for (1). Some assertions about instability in the framework of Lyapunov’s Direct Method for ordinary differential equations follow from the theorem (see [5]).