We consider stable linear scalar dynamical systems described by homogenous difference equations with nonzero initial conditions. We show that, at finite time instants, solutions can deviate far from the equilibrium. For classes of root locations and initial conditions, closed-form estimates are obtained for the magnitude of deviations and the instant of the maximum deviation. Simple conditions on the presence of deviations are formulated and the issues are discussed of how typical are these effects. Worst-case initial conditions yielding maximum peak are discussed. Several specific difference equations known from the literature are also analyzed and estimates of deviations are proposed. We also discuss multi-dimensional systems and provide upper bounds on deviations via use of linear matrix inequalities. The same technique is applied to the design of peak-minimizing feedback control laws.