Linear vs. Nonlinear

Abstract

Why do we need to have a nonlinear theory and why bother to study a qualitative nonlinear theory? After all, most models that are currently available are linear, and if a nonlinear model is to be used, computers are getting to be ever more powerful at simulating them. Do we really need a nonlinear theory? This is not a naive question, since linear models are so much more tractable than nonlinear ones and we can analyze quite sophisticated and high dimensional linear systems. Further, if one uses linear models with some possibly time-varying parameters, one may model real systems surprisingly well. Moreover, although nonlinear models may be conceptually more satisfying and elegant, they are of little use if one cannot learn anything from their behavior. Certainly, many practitioners in industry claim that they can do quite well with linear time varying models. Of course, an opposing argument is that we may use the ever increasing power of the computer to qualitatively understand the behavior of systems more completely and not have to approximate their behavior by linear systems.