A Generalization of the Theory of Normal Forms

Normal form theory is a technique for transforming the ordinary differential equations describing nonlinear dynamical systems into certain standard forms. Using a particular class of coordinate transformations, one can remove the inessential part of higher-order nonlinearities. Unlike the closely-related method of averaging, the standard development of normal form theory involves several technical assumptions about the allowed classes of coordinate transformations (often restricted to homogeneous polynomials). In a recent paper [1], the second author considered the equivalence of the methods of averaging and of normal forms. The references given there, particularly Chow and Hale [2], should be consulted for a full treatment of Lie Transforms. In this paper, we relax the restrictions on the transformations allowed. We start with the Duffing equation, and show that a singular coordinate transformation can remove the nonlinearity associated with the usual normal form. We give two interpretations of this coordinate transformation, one with a branch cut reminiscent of a Poincaré section. We then show, when the generating problem is linear and autonomous with diagonal Jordan form, that we can remove all nonlinearities order by order using singular coordinate transformations generated by the solution to the first-order linear partial differential equation produced by the Lie Transform method of normal form theory. A companion paper [4] discusses these methods in a more general context and treats a specific example with a nondiagonal Jordan form for the generating matrix.