Normal forms near critical points for differential equations and maps

The normal-form theory is a technique of transforming an original vector field to a simpler form by an appropriate change of coordinates, so that the essential features of the flow become more evident. A basic theory of normal forms, based on the classical idea of Poincare and Birkhoff, is presented. Normal forms for vector fields and diffeomorphisms are discussed, and their relationship is considered. The technique described is based on defining a certain linear operator and an inner product on the space of homogeneous polynomials on C ⁿ