There is no general theory of nonlinear systems. A nonlinear system with memory can be quite surprising. In particular, it can emit any output signal in response to any input signal. For example, it could replace all music by Beethoven with something by Mozart, etc. That said, many subclasses of nonlinear filters can be successfully analyzed:
One often-used tool for nonlinear systems analysis is Volterra series [4]. A Volterra series expansion represents a nonlinear system as a sum of iterated convolutions:
Here
Similarly, the third-order kernel
In the special case for which the Volterra expansion reduces to
we have an immediate frequency-domain interpretation in which the output spectrum is expressed as a power series in the input spectrum: