Normal Forms in Perturbation Theory

Abstract

Normal form procedure This is the stepwise ‘simplification’ by changes of coordinates, of the Taylor series at an equilibrium point, or of similar series at periodic or quasi-periodic solutions. Preservation of structure The normal form procedure is set up in such a way that all coordinate changes preserve a certain appropriate structure. This applies to the class of Hamiltonian or volume preserving systems, as well as to systems that are equivariant or reversible with respect to a symmetry group. In all cases the systems may also depend on parameters. Symmetry reduction The truncated normal form often exhibits a toroidal symmetry that can be factored out, thereby leading to a lower dimensional reduction. Perturbation theory The attempt to extend properties of the (possibly reduced) normal form truncation, to the full system.