Multiple Steady States, Limit Cycles and Chaotic Attractors in Evolutionary Games with Logit Dynamics

This paper investigates, by means of simple, three and four strategy games, the oc-currence of periodic and chaotic behaviour in a smooth version of the Best Response Dynamics, the Logit Dynamics. The main \u85nding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles, do occur under the Logit Dy-namics, even for three strategy games. Moreover, we show that the Logit Dynamics displays another bifurcation which cannot to occur under the Replicator Dynamics: the fold catastrophe. Finally, we \u85nd, in a four strategy game, a period-doubling route to chaotic dynamics under a weightedversion of the Logit Dynamics. JEL classi\u85cation: C72, C7