Nonlinear Modeling aims at a more accurate representation of physical reality where many systems are found to violate the basic prerequisite of linear models: the so-called superposition principle which states that the effect of the sum of multiple system inputs equals the sum of the effects of the individual inputs. This principle is often violated due to limitations in the maximum amplitude a physical quantity may reach, and due to basic physical laws which show a nonlinear relationship among the relevant variables of the problem domain. These nonlinear effects often become more prominent with the ongoing miniaturization of the electronic devices used for systems realization.
When choosing appropriate models, we can distinguish between physical or glassbox models and blackbox models. The former correspond to system descriptions in terms of parameterized equations and require substantial prior knowledge of the application domain. The latter rather provide universally approximating structures which are helpful to represent wide classes of nonlinear systems with minimum prior knowledge, which is compensated by the use of sophisticated model adaptation or learning algorithms.
Another distinction is between dynamical systems with fading memory, i.e., filter models and dynamical systems with non-fading memory, i.e., oscillator models. While the latter can exhibit intricate behavior when operated in a chaotic regime, the former are well described by so-called Volterra Series representations and by Reservoir Computing representations. We have worked on many aspects of these models such as nonlinear oscillator identification from measured data, a new theory of Volterra series for mixed continuous- and discrete-time dynamical systems, model complexity reduction, the application of Reservoir Computing to wireless sensor networks and many others.