The ability to store and manipulate information is a hallmark of computational systems. Whereas computers are carefully engineered to represent and perform mathematical operations on structured data, neurobiological systems perform analogous functions despite flexible organization and unstructured sensory input. Recent efforts have made progress in modeling the representation and recall of information in neural systems. However, precisely how neural systems learn to modify these representations remains far from understood. Here we demonstrate that a recurrent neural network (RNN) can learn to modify its representation of complex information using only examples, and we explain the associated learning mechanism with new theory. Specifically, we drive an RNN with examples of translated, linearly transformed, or pre-bifurcated time series from a chaotic Lorenz system, alongside an additional control signal that changes value for each example. By training the network to replicate the Lorenz inputs, it learns to autonomously evolve about a Lorenz-shaped manifold. Additionally, it learns to continuously interpolate and extrapolate the translation, transformation, and bifurcation of this representation far beyond the training data by changing the control signal. Finally, we provide a mechanism for how these computations are learned, and demonstrate that a single network can simultaneously learn multiple computations. Together, our results provide a simple but powerful mechanism by which an RNN can learn to manipulate internal representations of complex information, allowing for the principled study and precise design of RNNs.