From Averaging to Normal Forms

Abstract

The essence of the method of averaging is to use near-identity coordinate changes to simplify a system of differential equations. (This is clearly seen, for instance, in Section 2.9, where the original system is periodic in time and the simplified system is autonomous up to some order k.) The idea of simplification by near-identity transformations is useful in other circumstances as well. In the remaining chapters of this book we turn to the topic of normal forms for systems of differential equations near an equilibrium point (or rest point). This topic has much in common with the method of averaging. A slow and detailed treatment of normal forms, with full proofs, may be found in [203]. These proofs will not be repeated here, and they are not needed in concrete examples. Instead, we will survey the theory without proofs in this chapter, and then, in later chapters, turn to topics that are not covered in [203]. These include a detailed treatment of normal forms for Hamiltonian resonances and recent developments in the theory of higher-level normal forms. (By higherlevel normal forms we mean what various authors call higher-order normal forms, hypernormal forms, simplest normal forms, and unique normal forms.).