Using averaging methods of the theory of differential equations, the effects of the injection of high frequency signals, known as dithers, in non-linear systems is studied. The behaviour of the dithered system is compared with that of a smoothed one, whose non-linearity is the convolution of the dither distribution function and the original non-linearity. It is proved that if over a given time interval the smoothed system has a bounded solution, then so does the dithered system, provided that the dither frequency is high enough, and the output of the two systems can be made as close as desired. Moreover, if the trajectories of the smoothed system are asymptotically attracted to a compact set, for example a stable orbit or singular point, then the trajectories of the dithered system can be made to remain close to this set. In particular, a satisfactory explanation of quenching of limit cycldes is botained in this way. The analysis takes account of certain disturbances which might arise due to the injection of external signals. As an application of the theory, it is shown that with suitable dithers, and to within any degree of approximation, odd symmetric non. linearities which saturate can be made to behave like pure linear gains in many control systems. Finally, a comparison is made of these results with those of Zames and Schneydor (1976) in this area and the theory is illustrated by some examples.