General Principles for Nonlinear Systems

Abstract

Although nonlinear systems vary a great deal throughout science, their study is unified by fundamental mathematical principles. In this section we outline the simplest of these principles and illustrate just how they arise in practice. Of course, we saw in Chapter 1 how certain abstract principles were used in studying the simplest nonlinear systems, namely, integrable systems and their perturbations. These principles have the virtue of great “robustness” in the sense that they also have great value in studying more complicated nonlinear systems. In fact, one of the goals of this chapter is to illustrate just how the principles discussed briefly in Chapter 1 can be extended to study more general nonlinear systems such as occur throughout science and technology. Such systems are generally nonintegrable, even with our extended definition.