can arise in nature. For example, male side-blotched lizards [@Sin96] adopt one of three mating strategies: *ultra-dominant* with a large territory, *mate guarding* in a small territory or *sneaker* without territory, mating opportunistically. Dominant lizards beat guarders, but are vulnerable to sneakers, whereas guarders beat sneakers, creating a cyclical competition and oscillations in the frequencies of the three strategies. Cyclic competition can also arise in sociological contexts [@Sem03]. When the rock-scissors-paper game is studied using the replicator equations, the dynamics lack stable limit cycles. Depending on the values of payoffs for winning or losing, the system exhibits one of three kinds of behaviour: stable coexistence, neutrally stable cycles, or cycles of increasing amplitude [@Now06]. The first result we present is to show that in our memory based learning dynamics, limit cycles can form at critical memory length via a Hopf bifurcation [@Ern09]. The appearance of such cycles, also created by a Hopf bifurcation, has recently been discovered in the replicator equations, provided that mutations from one strategy to another [@Mob10], or more complex patterns of mutation [@Tou15] are allowed. In contrast to this work, due to the memory present in our system, our dynamics is most naturally described by delay 