In this paper, we construct a conjugacy between the time-1-map of the solution flow generated by an ordinary differential equation and its numerical approximation in a neighborhood of a fold bifurcation point. Our main result is that the conjugacy is O(h^p)-close to the identity on the center manifold where h is the step size and p is the order of the numerical method. © 2002 Elsevier Science Ltd. All rights reserved.