| Abstract: |
| Evolutionary game dynamics provide an explanation for how the collective behavior of a large population of players changes over time. This approach consists of two fundamental elements: a population game that describes the strategic interaction that is to occur repeatedly, and a replicator equation that outlines the procedure used by players to determine when and how to adopt new strategies. Drastically different dynamics of zero-sum and nonzero-sum games can be observed under replicator equations. In zero-sum games, heteroclinic cycles occur naturally when species of the population supersede each other in a cyclic fashion, like for the Rock-Paper-Scissors game. 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. F.Takens made a prediction in his work that certain dynamical systems found in evolutionary game theory and population dynamics could exhibit persistent historic behavior through the presence of attractive heteroclinic cycles. Recently, 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 talk, our objective is to confirm Takens` prediction by proposing a persistent and broad class of replicator equations which exhibit historic behavior wherein the slow oscillation of time averages of the orbit ultimately causes the divergence of higher-order repeated time averages. |
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