Real Birkhoff normal forms and complex coordinates
The author discusses the origins and the meaning of the concept of a real Birkhoff normal form of a vector field near a non-hyperbolic equilibrium point and works out a method for the construction of normal forms in a complex framework. He shows how to get an information on a real normal form problem through an appropriate translation into an equivalent complex problem. Starting with an embedding of a real vector space into a complex one he constructs the respective embedding of vector fields and the analogue of the so-called homological operator. The paper offers a systematic treatment of the technique of introducing complex coordinates, shows how these transformations are computed and executed and how standard normal forms may be characterized in the complex formalism. Some important special cases are carefully studied providing explicit formulas and some well-known examples are discussed.
scientific article; zbMATH DE number 168467