A Generalization of the Theory of Normal Forms
chao-dyn
2009-10-28 v1 Chaotic Dynamics
Abstract
Normal forms allow the use of a restricted class of coordinate transformations (typically homogeneous polynomials) to put the bifurcations found in nonlinear dynamical systems into a few standard forms. We investigate here the consequences of relaxing the restrictions of the form of the coordinate transformations. In the Duffing equation, a logarithmic transformation can remove the nonlinearity: in one interpretation, the nonlinearity is replaced by a branch cut leading to a Poincare section. When the linearized problem is autonomous with diagonal Jordan form, we can remove all nonlinearities order by order using these singular coordinate transformations.
Cite
@article{arxiv.chao-dyn/9510014,
title = {A Generalization of the Theory of Normal Forms},
author = {W. H. Warner and P. R. Sethna and James P. Sethna},
journal= {arXiv preprint arXiv:chao-dyn/9510014},
year = {2009}
}