It is known that the one-dimensional discrete maps having single-humped nonlinear functions with the same order of maximum belong to a single class that shows the universal behaviour of a cascade of period-doubling bifurcations from stability to chaos with the change of parameters. This paper concerns studies of the dynamics exhibited by some of these simple one-dimensional maps under constant perturbations. We show that the “universality” in their dynamics breaks down under constant perturbations with the logistic map showing different dynamics compared to the other maps. Thus these maps can be classified into two types with respect to their response to constant perturbations. Unidimensional discrete maps are interchangeably used as models for specific processes in many disciplines due to the similarity in their dynamics. These results prove that the differences in their behaviour under perturbations need to be taken into consideration before using them for modelling any real process.