Transient performance problems in stable linear systems were always in focus of the control community. Nevertheless, deviations of trajectories in input-free discrete time systems have been paid much less attention. In this paper, we are interested in estimating the magnitudes of deviations caused by nonzero initial conditions. First, we present examples where the magnitude of peak can be computed in closed form and show that it may take very large values. Next, we propose numerical procedures for the computation of upper bounds on deviations and for the design of peak-attenuating controllers. Finally, we present an extension to the case of uncertain systems. Numerical experiments confirm the efficiency of the approach and demonstrate its low conservatism.