||Symmetry has been a major ingredient in the development of quantum perturbation theory, and it is a basic ingredient of the theory of integrable (Hamiltonian and non-Hamiltonian) systems; yet, the use of symmetry in the context of general perturbation theory is rather recent. Prom the point of view of nonlinear dynamics, the use of symmetry has become widespread only through equivariant bifurcation theory; even in this case, attention has been mostly confined to linear symmetries.
Also, in recent years the theory and practice of symmetry methods for differential equations has become increasingly popular and has been applied to a variety of problems (following the appearance of the book by Olver ). This theory is deeply geometrical and deals with symmetries of general nature (provided that they are described by vector fields), i.e. in this context there is no reason to limit attention to linear symmetries.
In this book we look at the basic tools of perturbation theory, i.e. normal forms (first introduced by Poincare about a century ago) and study their interaction with symmetries, with no limitation to linear ones. We focus on the most basic setting, i.e. systems having a fixed point (at the origin) and perturbative expansions around this; thus our theory is entirely local.
We have tried to give a reasonably self-contained discussion, so the first three chapters deal with symmetry, differential equations, and in particular dynamical systems in a quite general way (and with many exercises to help the beginner). This part represents a compact introduction to symmetric dynamical systems and does not contain new results.
In Chaps. IV and V we discuss normal forms in the presence of symmetries, both in the general (dynamical systems) case and in the Hamiltonian one, at the formal level, i.e. without considering the convergence of the power series entering in the theory. This convergence is studied in Chap. VI, again focusing on the role of symmetries. Many of the results presented in this part are quite recent, having been obtained in the last five years (mostly between 1994 and 1997), and some of them are actually completely new.
In the final part of the book we discuss some related problems. In Chap. VII we discuss the relation between symmetries of a dynamical system and its invariant manifolds, in particular the center manifold; in Chap. VIII we discuss an extension of normal forms theory, i.e. the "further normalization" of normal forms; and finally in Chap. IX we discuss a formal approach to asymptotic symmetries.