You make a good point. Indeed a transfer function is usually used to characterise linear systems. However, non-linear systems also have transfer functions, it

You make a good point. Indeed a transfer function is usually used to characterise linear systems. However, non-linear systems also have transfer functions, it's just that they are not scale invariant. Transfer function technically is just the "function" characterising the relationship between input and output. In a time-invariant linear system this is the ratio of the input spectrum to output spectrum, and is invariant to the scale of the input, and so can be characterised by a matrix multiplication (ie a linear transformation). For non-linear systems such as distortion, you now have a time-varying linear transfer function, a gain, depending on the amplitude of the input. So overall it is non-linear, and must be characterised by a function (the distortion curve), not by a single linear operation. One of the consequences of this is that it's a lot harder to go between the time domain and the frequency domain when dealing with non-linear transformations, so frequency analysis becomes more difficult -- but not impossible, depending on the system. For instance Chebyshev distortion can be analysed in the frequency domain. So, non-linear transfer functions can be seen as arbitrary functions, or can be seen as linear transfer functions parameterised by some parameter that varies non-linearly with the input (eg a gain modulated by a non-linear function of the input amplitude -- distortion). This second view is called "linearization" where you model a non-linear system with a local linear model, based on the assumption that non-linearities are smooth when viewed "close up", and is needed to apply certain linear control methods to non-linear systems.