A statistical non linearization method is used to approximate systems modeled by the Bouc differential equation and excited by a Gaussian white noise external load. To this aim restricted potential models (RPM) are used, which are suitable for an extended number of nonlinear problems as have been proved several times. Since the solution of RPM is known by the probabilistic point of view, all statistical characteristics can be derived at once with advantages by the computational point of view. Hence, this paper discusses the possibility to determine sets of parameters characterizing potential models that are valid for describing a hysteretic behavior. In this way the characterization of the hysteretic behavior of a system can be performed with computational efforts lower than that normally requested.