Even though existence of non-convergent evolution of the states of populations in ecological and evolutionary contexts is an undeniable fact, insightful game-theoretic interpretations of such outcomes are scarce in the literature of evolutionary game theory. As a proof-of-concept, we tap into the information-theoretic concept of relative entropy in order to construct a game-theoretic interpretation for periodic orbits in a wide class of deterministic discrete-time evolutionary game dynamics, primarily investigating the two-player two-strategy case. Effectively, we present a consistent generalization of the evolutionarily stable strategy -- the cornerstone of the evolutionary game theory -- and aptly term the generalized concept: information stable orbit. The information stable orbit captures the essence of the evolutionarily stable strategy in that it compares the total payoff obtained against an evolving mutant with the total payoff that the mutant gets while playing against itself. Furthermore, we discuss the connection of the information stable orbit with the dynamical stability of the corresponding periodic orbit.