Equivariant decomposition of polynomial vector fields

Fahimeh Mokhtari, J.A. Sanders

Research output: Contribution to JournalArticleAcademicpeer-review


To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the this paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch–Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D cases, followed by an application to the computation of the unique normal form of the Rössler equation.
Original languageEnglish
Article number2050083
Number of pages31
JournalCommunications in Contemporary Mathematics
Issue number8
Early online date23 Dec 2020
Publication statusPublished - Dec 2021


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