Computational aspects of normal form expansions

We use the method of normal forms (NF) to explore perturbative solutions to a broad class of nonlinear differential equations and to symplectic planar maps. The method employs introduction of a transformation from the original variables to a new set that yields equations that are more amenable to analysis. The transformation is not unique and ‘‘freedom’’ is thereby introduced, which is exploited to tailor the structure of the expansion to match various constraints that one might impose. Kahn and Zarmi, and Kahn, Murray, and Zarmi have introduced and used the method of minimal normal forms (MNF) and variations thereof that exploit the lack of uniqueness and consequent freedom of choice in the normal form expansion to realize ‘‘compact’’ expansions that have a simple structure, are easy to use, and have significant computational advantages over other schemes.