Dynamical system generated by a nonlinear operator acting on the real line or an ordered metric space and having increasing trajectories is considered. Such an operator is said to be mapping (operator), majorized below by identity map. The effect of changing the generating operator by the operator with lesser or greater values is studied. The main results of the work are the following: (a) For the composition of generating mappings the sufficient condition to reserve the property of the stabilization of all system trajectories is obtained; (b) It is proved that if the set of all unbounded trajectories of the operator acting on the real line is not empty, than any greater operator has the same property (with respect to the pointwise order relation). (c) It is proved that there are systems on real line with increasing trajectories, such that we may change their generating operators by arbitrary close ones (by subtracting small constants from their values), and obtain the systems having hidden attractors (in the sense of N.Kuznetsov). The example illustrating (c) is given.