On the response of physical systems governed by non-linear ordinary differential equations to step input

Abstract

The response of physical systems governed by linear ordinary differential equations to step input is traditionally investigated using the classical theory of distributions. The response of non-linear systems is however beyond the reach of the classical theory. The reason is that the simplest non-linear operation—multiplication—is not defined for distributions. Yet the response of non-linear systems is of interest in many applications, and a framework capable of handling such problems is needed.We argue that a suitable framework is provided by Colombeau algebra that gives one the possibility to overcome the limitations of the classical theory of distributions, namely the possibility to simultaneously handle discontinuity, differentiation and non-linearity. Our thesis is documented by means of studying the response of two systems governed by non-linear ordinary differential equations subject to step input. In particular, we show that using the rules of calculus in Colombeau algebra it is possible to obtain an explicit and practically relevant characterisation of the behaviour of the considered systems at the point of the jump discontinuity.