A general method is presented for determining the bounds on allowable disturbances, in linearly stable systems, for which the system remains asymptotically stable. It is based on transforming a set of nonlinear differential equations to a single equation that is valid within a given region of equilibrium. It is applicable to systems with a fairly general nonlinear feedback as well as to systems that exhibit finite escape time, thus extending previous methods. The physics enters through the linear characteristic roots, and provision is made for both real and complex roots. The method is also of use in determining the range of validity of space-independent reactor models. Applications are given to three examples of reactor systems, including the determination of reactor excursions.