A dynamical system approach to forward and inverse modeling is proposed. Forward and inverse models are trained in associative recurrent neural networks that are based on non-linear random projections. Feedback of estimated outputs into such reservoir networks is a key ingredient in the context of bidirectional association but entails the problem of error amplification. Robust training of reservoir networks with output feedback is achieved by a novel one-shot learning and regularization method for input-driven recurrent neural networks. It is shown that output feedback enables the implementation of ambiguous inverse models by means of multi-stable dynamics. The proposed methodology is applied to movement generation of robotic manipulators in a feedforward-feedback control framework.