Normal forms for nonlinear vector fields. I. Theory and algorithm

Normal forms are analytical tools for studying the qualitative behavior of the nonlinear vector fields. A tutorial for the nonspecialist in general, and the circuit theorist in particular, on the basic concept and foundation of the modern theory of normal forms for nonlinear vector fields, is presented. After stating the Poincare and the Takens normal form, the latest refinements due to S. Ushiki (1984) are pointed out. For pedagogical reasons, the familiar Jordon form is first derived and shown to be appropriate normal form for matrices. Rather than using a standard linear approach, formulation is based on the `method of infinitesimal deformation´ which generalizes to nonlinear vector fields