16 - Normal Forms: Near-Identity Transformations

Abstract

This chapter elaborates normal forms for near-identity transformations. The concept of normal forms does not require the transformations used be near-identity ones; but they are the ones most often used in practice. For getting the normal form, the idea is to find a change of variables, in the form of an infinite series, so that the original system of differential equations goes into a “normal” (or “simple” or “canonical”) form. The normal form is the simplest member of an equivalence class of differential equations, all exhibiting the same qualitative behavior. Normal forms are often useful for stability analyses. The procedure starts with the system x' = f(x) such that (without loss of generality) x = 0 is a critical point. This system is expanded to obtain x' = Ax + H(x), where H(x) has strictly nonlinear functions. If the critical point is “hyperbolic” (all eigenvalues have nonzero real parts) then the nonlinear terms can always be removed.