Nonlinear systems and exponential eigenfunctions

It has been shown previously that homogeneous time-invariant systems produce exponential outputs in response to similar exponential inputs, but that the concepts of "impulse response" and "frequency response" are of little use for their analysis. It was also asked whether there exist more general classes of systems with exponential eigenfunctions. In this paper, we recall that the concepts of impulse and frequency response can be useless even for certain linear, time-invariant systems. We discuss the role of time-invariance, commuting linear systems, and conditions under which they have common eigenfunctions. Then we exhibit a class of nonlinear, nonhomogeneous, time-varying systems that still have exponential eigenfunctions. This class contains homogeneous time-invariant systems, finite impulse response (FIR) filters and generalized feedforward filters as special cases, and it shows that the exponential eigenfunction property does not imply linearity, homogeneity, or time-invariance.