A Generalization of the Theory of Normal Forms

Details A Generalization of the Theory of Normal Forms A Generalization of the Theory of Normal Forms

1995-10-23

arxiv.org/abs/chao-dyn/9510014v1

Nonlinear Sciences

Chaotic Dynamics

Scientific paper

10.1007/BF02434054

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.

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