The surfaces of alluvial fans and river deltas (collectively fans) are often dissected by a small number of channels radiating from the fan apex. On long timescales, channels migrate via avulsion, the process of channel bed deposition and abandonment that often results in catastrophic flooding and loss of life on densely populated fans. We present results of an experimental fan that creates realistic channel patterns by avulsion. The avulsion cycle occurs with a period that is predictable from conservation of mass. Selection of a new flow path is inherently stochastic; however, once a network of 4-5 channels is established, flow oscillates among these channels indefinitely. We demonstrate that a directed random walk model with memory quantitatively reproduces these dynamics and limiting behavior, and is consistent with natural fans.