An explicit mapping between the frequency domain and the time domain representations of non-linear systems

Explicit expressions are presented that describe the input-output behaviour of a non-linear system in both the frequency and the time domain. The expressions are based on a set of coefficients that do not depend on the input to the system and are universal for a given system. The anharmonic oscillator is chosen as an example and is discussed for different choices of its physical parameters. It is shown that the typical approach for the determination of the Volterra Series representation is not valid for the important case when the non-linear system exhibits oscillatory behaviour and the input has a pole at the origin (in the frequency domain), e.g. the unit-step function. For this case, resonant effects arise and the analysis requires additional care.