  In evolutionary game theory, it is customary to be partial to the dynamical
models possessing fixed points so that they may be understood as the attainment
of evolutionary stability, and hence, Nash equilibrium. Any show of periodic or
chaotic solution is many a time perceived as a shortcoming of the corresponding
game dynamic because (Nash) equilibrium play is supposed to be robust and
persistent behaviour, and any other behaviour in nature is deemed transient.
Consequently, there is a lack of attempt to connect the non-fixed point
solutions with the game theoretic concepts. Here we provide a way to render
game theoretic meaning to periodic solutions. To this end, we consider a
replicator map that models Darwinian selection mechanism in unstructured
infinite-sized population whose individuals reproduce asexually forming
non-overlapping generations. This is one of the simplest evolutionary game
dynamic that exhibits periodic solutions giving way to chaotic solutions (as
parameters related to reproductive fitness change) and also obeys the folk
theorems connecting fixed point solutions with Nash equilibrium. Interestingly,
we find that a modified Darwinian fitness -- termed heterogeneity payoff -- in
the corresponding population game must be put forward as (conventional) fitness
times the probability that two arbitrarily chosen individuals of the population
adopt two different strategies. The evolutionary dynamics proceeds as if the
individuals optimize the heterogeneity payoff to reach an evolutionarily stable
orbit, should it exist. We rigorously prove that a locally asymptotically
stable period orbit must be heterogeneity stable orbit -- a generalization of
evolutionarily stable state.
