Amplitudes and stability of the harmonic oscillations in closed-loop systems with ‘ dynamic range ’ type non-linearity under sudden sinusoidal excitation are analysed using the describing function mid the phase-plane methods. The legions of initial conditions leading to a particular amplitude of oscillation are also determined. From this a general and qualitative behaviour of the response and also the nature of variation of the multiple amplitudes in systems containing a piece-wise, single-valued non-linear characteristic are discussed. The study is repeated for the case in which the input amplitude is varied at a fixed frequency. Analogue computer results, for one example, show the correspondence with the theory. The method is then extended to systems with backlash.