On difference equations with asymptotically stable 2-cycles perturbed by a decaying noise

The results stating that the stability of a 2-cycle is preserved (almost surely) under an eventually decaying stochastic perturbation are obtained in the case when the system is in the range of the parameters immediately following the Hopf bifurcation and preceding the next period doubling bifurcation. Several examples of systems and types of noise are presented.