A GENERALIZATION OF THE THEORY OF NORMAL FORMS

Normal forms allow the use of a restricted class of coordinate transfo rmations (typically homogeneous polynomials) to put the bifurcations f ound in nonlinear dynamical systems into a few standard forms. We inve stigate here the consequences of relaxing the restrictions of the form of the coordinate transformations. In the Duffing equation, a logarit hmic transformation can remove the nonlinearity: in one interpretation , the nonlinearity is replaced by a branch cut leading to a Poincare s ection. When the linearized problem is autonomous with diagonal Jordan form, we can remove all nonlinearities order by order using these sin gular coordinate transformations.