Reduction of linear differential systems to Arnold canonical form

Abstract

We consider systems of linear differential equations of the form ε^Pdy/dx=A(k,ε)y where A is an n×n matrix-valued function holomorphic in x possessing an asymptotic power series expansion in ε, A(k,ε)≈Σ_kA_k(x)ε^k uniformly valid on a neighborhood |x|<r of x=0 as ε→0⁺. We show that any such system can be reduced by a change of variable y→Ty to a certain generic canonical form depending on the Jordan canonical form of A(0, 0). Here T(x, ε) is an invertible matrix-valued function similarly possessing an asymptotic expansion T≈Σ_kT_k(x)ε^k uniformly valid on an ε-dependent domain |x|<rδ(ε) the size of which is determined by certain finer properties of the asymptotic expansion of A. We also obtain new results of classical type in which this domain is independent of ε. The method of proof seems both new and of general purpose. It is based on an exacting analysis of finer termwise properties of divergent infinite formal series transformations which shows certain near-optimal finite sections to be powerful first approximations to actual transformations.