Center manifolds of dynamical systems under discretization

We consider an autonomous dynamical system discretized by a one-step method. The point z = 0 is assumed to be fixed under the continuous and the discrete flows. We allow z = 0 to be non-hyperbolic. The continuous system has a center-unstable manifold and we show the existence of approximating invariant manifolds for the discretizations. The manifolds for the continuous and the discrete systems share the property of being locally attracting at an exponential rate; the dynamics inside the manifolds can differ qualitatively, however, for all step-sizes h.