DISCRETIZATION IN THE METHOD OF AVERAGING

Let f: R x R(m)BAR x R --> R(m)BAR, f = f(epsilon, x, t) be a C2-mapping 1-periodic in t having the form f(0, x, t) = Ax + o(\x\) as x --> 0 where A is-an-element-of L (R(m)BAR) has no eigenvalues with zero real parts. We study the relation between local stable manifolds of the equation x' = epsilon . f(epsilon, x, t), epsilon > 0 is small and of its discretization x(n + 1) = x(n) + (epsilon/m) . f(epsilon, x(n), t(n)), t(n + 1) = t(n) + 1/m, where m is-an-element-of {1, 2, ...} = N. We show behavior of these manifolds of the discretization for the following cases: (a) m --> infinity, epsilon --> epsilonBAR > 0, (b) m --> infinity, epsilon --> 0, (c) m --> k is-an-element-of N, epsilon --> 0.