The flow of droplets through the simplest microfluidic network—a set of two parallel channels with a common inlet and a common outlet—exhibits a rich variety of dynamic behaviors parametrized by the frequency of feeding of droplets into the system and by the asymmetry of the arms of the microfluidic loop. Finite ranges of these two parameters form islands of regular (cyclic) behaviors of a well defined period that can be estimated via simple theoretical arguments. These islands are separated by regions of behaviors that are either irregular or cyclic with a very long periodicity. Interestingly, theoretical arguments and numerical simulations show that within the islands of regular behaviors the state of the system can be degenerate: there can exist a number of distinct sequences of trajectories of droplets, each stable and—in the absence of disturbances—continuing ad infinitum. The system can be switched between these cyclic trajectories with a single stimulus.