Artificial neural networks (ANNs) are often used for short term discrete time predictions of experimental data. In this paper we focus on the capability of such nets to correctly identify long term behavior and, in particular, observed bifurcations. As we show, the usual discrete time mapping approach is (precisely because of its discrete nature) often incapable of reproducing observed bifurcation sequences. If the interest is only in periodic or temporally more complicated behavior, a Poincare map extracted from the experimental time series can be used to circumvent this problem. A complete dynamic picture including bifurcations of steady states can, however, only be captured by a continuous-time model. We present an ANN configuration which couples a ''nonlinear principal component'' network for data processing (Kramer, 1991, Usui et al., 1990) with a composite ANN based on a simple integrator scheme. This ANN is able to correctly reconstruct the bifurcation diagram of our experimental data. All time series we process stem from the potentiostatic electrodissolution of Cu in phosphoric acid solution. As the applied potential is varied, the electrodissolution rate changes from steady behavior to periodic oscillations, followed by a sequence of period doublings to apparently chaotic motion, and then returns to simple oscillations via a reverse cascade of period doublings.