Algorithmic Reduction of Poincaré–Dulac Normal Forms and Lie Algebraic Structure

The Poincaré–Dulac normal form of a given resonant system is in general nonunique; given a specific normal form, one would like to further reduce it to a simplest normal form. In this Letter we give an algorithm, based on the Lie algebraic structure of the set of normal forms, to obtain this. The algorithm can be applied under some condition, nongeneric but often met in applications. When applicable, it is only necessary to solve linear equations, and is more powerful than the one proposed in previous work by the same author [Lett. Math. Phys. 42 (1999), 103–114; and Ann. Inst. H. PoincaréPhys. Théor. 70 (1999), 461–514].