Normalizations with exponentially small remainders for nonautonomous analytic periodic vector fields

TL;DR

This paper extends normal form theory to nonautonomous periodic vector fields, demonstrating that under certain conditions, one can achieve exponentially small remainders through polynomial coordinate changes, generalizing autonomous case results.

Contribution

It introduces a normal form theorem with exponentially small remainders for nonautonomous periodic vector fields, generalizing previous autonomous case results.

Findings

Existence of polynomial coordinate changes eliminating certain terms

Achieving exponentially small remainders in normal forms

Generalization of autonomous case theorems to nonautonomous periodic systems

Abstract

In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependancy, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E0 and E1. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E1 allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u0 of E0 in the E1 component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case : this paper generalizes those results to the nonautonomous periodic case.