Summary
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 Poincaré 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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Communicated by Stephen Wiggins
Deceased, November 4, 1993.
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Warner, W.H., Sethna, P.R. & Sethna, J.P. A generalization of the theory of normal forms. J Nonlinear Sci 6, 499–506 (1996). https://doi.org/10.1007/BF02434054
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DOI: https://doi.org/10.1007/BF02434054