Some aspects of the theory of nonlinear networks

Abstract

In the development of network theory over the years, the primary focus of attention has been in the area of linear systems. Several reasons for this emphasis can easily be cited, but perhaps the foremost reason is that it has long been thought that, except in certain very special cases, little progress toward a rigorous definitive theory could be expected once the hypothesis of linearity is discarded. The recent success in the use of numerical methods for computing solutions of the equations for specific nonlinear networks (the importance of which is not to be minimized) has, furthermore, resulted in a certain complacency on the part of many engineers who occasionally need to solve network problems. One senses their outlook as being, basically, that whenever a particular nonlinear problem arises, one need only then run, data in hand, to the computer. Somewhat ironically, however, the development of computer-aided network analysis techniques has also been a prime impetus for many of the recent theoretical investigations in the field of nonlinear networks, and although much remains to be done, a rather comprehensive body of knowledge in this area has begun to take form. A number of related recent contributions to the theory of non-linear networks are reviewed here. As distinct from the computational aspects of the network analysis problem, we discuss work whose primary purpose is to yield an understanding of the nature of the equations that describe the behavior of nonlinear networks, and to identify and relate certain properties of the network elements, and the manner of their interconnection, to properties of the equations and their solutions. In addition, we do frequently touch on the problem of computation since, as has already been implied, it is indeed one of the purposes of the work discussed here to provide more of a theoretical foundation on which to base the numerical analyses.