Abstract It is shown how a frequency domain criterion can be used to obtain meaningful bounds on the responses of nonlinear feedback systems. If the system is disturbed by a known input or nonzero initial condition from its state of equilibrium, bounds are obtained on overshoot and settling time as the system returns to its state of equilibrium. The method presented also permits to compute bounds for responses to bounded inputs which do not bring the system to a state of equilibrium. The bounds have a graphical interpretation in the Nyquist-plane which is similar in concept to M-circles for linear systems. This graphical interpretation can be used advantageously in the design nonlinear feedback systems.