Historic Behavior in Rock–Paper–Scissor Dynamics

In the evolutionary dynamics of the Rock–Paper–Scissor game, it is common to see the emergence of heteroclinic cycles. The dynamics in the vicinity of a stable heteroclinic cycle is marked by intermittency, where an orbit remains close to the heteroclinic cycle, repeatedly approaching and lingering at the saddles for increasing periods of time, and quickly transitioning from one saddle to the next. This causes the time spent near each saddle to increase at an exponential rate. This highly erratic behavior causes the time averages of the orbit to diverge, a phenomenon known as historic behavior. The problem of describing persistent families of systems exhibiting historic behavior, known as Takens’ Last Problem, has been widely studied in the literature. In this paper, we propose a persistent and broad class of replicator equations generated by increasing functions which exhibit historic behavior wherein the slow oscillation of time averages of the orbit ultimately causes the divergence of higher-order repeated time averages.