Do drainage basins develop static river networks when subject to steady forcing? While current landscape evolution models differ in formulation and implementation, they have the common characteristic that when run for long times at constant forcing, they evolve to a static steady-state configuration in which erosion everywhere balances uplift rate. This results in temporally stationary ridge and valley networks. We have constructed a physical model of a drainage basin in which we can impose constant rainfall and uplift conditions. The model landscapes never become static, and they are not sensitive to initial surface conditions. Ridges migrate laterally, change length, and undergo topographic inversion (streams occupy former ridge locations). Lateral stream migration can also produce strath terraces. This occurs without any change in external forcing, so the terraces must be considered autocyclic. The experimental drainage basin also exhibits autocyclic (internally generated) oscillations in erosion rate over a variety of time scales, despite constant forcing. The experimental landforms are clearly not perfect analogs of natural erosional networks, but the results raise the possibility that natural systems may be more dynamic than the current models would suggest, and that features like strath terraces that are generally interpreted in terms of external forcing may arise autocyclically as well.