Linearization & Transformation

Abstract

In this chapter we present two basic techniques that are useful for analyzing nonlinear systems. One technique consists of linearizing about the fixed points to obtain local, qualitative pictures of the phase portrait via the corresponding linear systems. Thus, our previous work on linear systems has direct bearing on nonlinear systems. The validity of this technique is contained in the Linearization Theorem, which we present later on, after first applying it in numerous examples and exercises.

The other technique studied here, which has broader significance, involves the idea of transforming one system of DEs into another, perhaps simpler, system. You have already studied this for linear systems where the transformation was via a linear transformation and the resulting simpler system was the linear system determined by the Jordan form. The general technique uses nonlinear transformations and is motivated by the example of transforming to polar coordinates which you studied in Chapter 2. The transformation theory also motivates the notion of topological equivalence of systems of DEs, which is the basis for the Linearization Theorem.