The investigation is performed by taking as a model an electrical circuit (a resonant parallel circuit) but the results could be extended by analogy to the case of a mechanical system. The simple hysteresis model proposed by Hayashi has been used in the case of sinusoidal excitation; this model allows one to write a differential equation which admits of an integrable Fokker-Planck equation. Various statistical quantities are evaluated and are compared with the results given by another method. The principal conclusions are as follows: the evidence with regards to the non-linear oscillator without hysteresis is that the response is always very feeble; in the case where the excitation consists of a sinusoidal signal and Gaussian white noise, the evidence is that the ratio of the useful part of l the response to that of the noise is improved. UR - https://www.ingentaconnect.com/content/dav/aaua/1966/00000017/00000006/art00007 ER -