Before revision:  The most striking application of periodicity is that in mixed strategy strategic form games, we were able to find solutions that result to values for the utility function of each player , that are equal to the Nash equilibrium ones , with the difference that in the Nash strategiesplaying, the payoffs strongly depend on what the opponent plays, while in the periodic strategies case, the payoffs of each player are completely robust against what the opponent plays.
After revision:  The most striking application of periodicity is that in mixed strategy strategic form games, periodic solutions yield values for the utility function of each player , that are equal to the Nash equilibrium ones , with the difference that in the Nash strategiesplaying, the payoffs strongly depend on what the opponent plays, while in the periodic strategies case, the payoffs of each player are completely robust against what the opponent plays.